CBD graphs

Manuscript/sfBSE.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2021-01-13 09:41:10 +0100 
+%% Created for Pierre-Francois Loos at 2021-01-14 16:29:49 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -1017,10 +1017,12 @@
 @article{Loos_2020a,
 	author = {P. F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse},
 	date-added = {2020-12-09 09:59:26 +0100},
-	date-modified = {2020-12-09 09:59:26 +0100},
+	date-modified = {2021-01-14 16:29:23 +0100},
+	doi = {10.1021/acs.jctc.9b01067},
 	journal = {J. Chem. Theory Comput.},
-	pages = {in press},
+	pages = {1018--1028},
 	title = {A Density-Based Basis-Set Incompleteness Correction for GW Methods},
+	volume = {16},
 	year = {2020},
 	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01103}}
 

Manuscript/sfBSE.tex

@@ -610,19 +610,19 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f
 		\begin{tabular}{lcccc}
 									&	\mc{4}{c}{Excitation energies (eV)}			\\
 									\cline{2-5}
-			Method					&	$^3P(1s^22s2p)$	&	$^1P(1s^22s2p)$	
-									&	$^3P(1s^22p^2)$	&	$^1D(1s^22p^2)$			\\
+			Method					&	$^3P(1s^2 2s^1 2p^1)$	&	$^1P(1s^2 2s^1 2p^1)$	
+									&	$^3P(1s^22 p^2)$		&	$^1D(1s^22p^2)$			\\
 			\hline
 			SF-TD-BLYP\fnm[1]		&	3.210	&	3.210	&	6.691	&	7.598	\\
 			SF-TD-B3LYP\fnm[1]		&	3.332	&	4.275	&	6.864	&	7.762	\\	
 			SF-TD-BH\&HLYP\fnm[1]	&	2.874	&	4.922	&	7.112	&	8.188	\\		
-			SF-BSE@{\GOWO}\fnm[2]	&	2.399	&	6.191	&	7.792	&	9.373	\\
-			SF-BSE@{\evGW}\fnm[2]	&	2.407	&	6.199	&	7.788	&	9.388	\\
-			SF-BSE@{\qsGW}\fnm[2]	&	2.376	&	6.241	&	7.668	&	9.417	\\
-			SF-dBSE@{\GOWO}\fnm[2]	&	2.363	&	6.263	&	7.824	&	9.424	\\
-			SF-dBSE@{\evGW}\fnm[2]	&	2.369	&	6.273	&	7.820	&	9.441 	\\
-			SF-dBSE@{\qsGW}\fnm[2]	&	2.335	&	6.317	&	7.689	&	9.470	\\
-			SF-CIS\fnm[3]			&	2.111	&	6.036	&	7.480	&	8.945	\\
+			SF-CIS\fnm[2]			&	2.111	&	6.036	&	7.480	&	8.945	\\
+			SF-BSE@{\GOWO}\fnm[3]	&	2.399	&	6.191	&	7.792	&	9.373	\\
+			SF-BSE@{\evGW}\fnm[3]	&	2.407	&	6.199	&	7.788	&	9.388	\\
+			SF-BSE@{\qsGW}\fnm[3]	&	2.376	&	6.241	&	7.668	&	9.417	\\
+			SF-dBSE@{\GOWO}\fnm[3]	&	2.363	&	6.263	&	7.824	&	9.424	\\
+			SF-dBSE@{\evGW}\fnm[3]	&	2.369	&	6.273	&	7.820	&	9.441 	\\
+			SF-dBSE@{\qsGW}\fnm[3]	&	2.335	&	6.317	&	7.689	&	9.470	\\
 			SF-ADC(2)-s\fnm[2]		&	2.433	&	6.255	&	7.745	&	9.047 	\\
 			SF-ADC(2)-x\fnm[2]		&	2.866	&	6.581	&	7.664	&	8.612	\\
 			SF-ADC(3)\fnm[2]		&	2.863	&	6.579	&	7.658	&	8.618 	\\
@@ -630,13 +630,13 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f
 		\end{tabular}
 	\end{ruledtabular}
 	\fnt[1]{Value from Ref.~\onlinecite{Casanova_2020}.}
-	\fnt[2]{This work.}
-	\fnt[3]{Value from Ref.~\onlinecite{Krylov_2001a}.}
+	\fnt[2]{Value from Ref.~\onlinecite{Krylov_2001a}.}
+	\fnt[3]{This work.}
 \end{table}
 \end{squeezetable}
 %%% %%% %%% %%%
 
-%%% FIG. 1 %%%
+%%% FIG 1 %%%
 \begin{figure}
 	\includegraphics[width=\linewidth]{Be}
 	\caption{
@@ -645,6 +645,7 @@ In the left part of Fig.~\ref{fig:Be} we have results for the SF-TD-DFT, where f
 	All the spin-flip calculations have been performed with a UHF reference.
 	\label{fig:Be}}
 \end{figure}
+%%% %%% %%% %%%
 
 %%% TABLE II %%%
 %\begin{squeezetable}
@@ -761,18 +762,17 @@ Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and wi
 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 we can observe that for all the methods that we compared, when the spin-flip is not used standard methods can not retrieve double excitation. There is no avoided crossing or perturbation in the curve for the E${}^1 \Sigma_g^+$ state when spin-flip is not used. This is because for these methods we are in the space of single excitation and de-excitation.
 
 
+%%% FIG 2 %%%
 \begin{figure}
-\includegraphics[width=1\linewidth]{H2_CIS.pdf}
-\includegraphics[width=1\linewidth]{H2_BHHLYP.pdf}
-\includegraphics[width=1\linewidth]{H2_BSE.pdf}
+	\includegraphics[width=1\linewidth]{H2_CIS.pdf}
+	\includegraphics[width=1\linewidth]{H2_BHHLYP.pdf}
+	\includegraphics[width=1\linewidth]{H2_BSE.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 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}
-
-
-
+%%% %%% %%% 
 
 %===============================
 \subsection{Cyclobutadiene}
@@ -793,11 +793,24 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
 Table~\ref{tab:CBD_D2h} shows results obtained for the $D_{2h}$ rectangular equilibrium geometry and Table~\ref{tab:CBD_D4h} shows results obtained for $D_{4h}$ square equilibrium geometry. These results are given with respect to the singlet ground state. For each geometry three states are under investigation, for the $D_{2h}$ CBD we look at the $1 ~^3 B_{1g}$, $1~^1 B_{1g}$ and $2 ~^1 A_{1g}$ states. For the $D_{4h}$ CBD we look at the $1 ~^3 A_{2g}$, $2~^1 A_{1g}$ and $1 ~^1 B_{2g}$ states. Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections. In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, i.e., SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state. This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state. Adding dynamical corrections in SF-dBSE@{\GOWO} do not improve the accuracy of the excitation energies comparing to SF-ADC(3). Indeed, we have a difference of 0.052 eV for the $1 ~^3 B_{1g}$ state, 0.393 eV for the $1 ~^1 B_{1g}$ state and 0.293 eV for the $2 ~^1 A_{1g}$ state.
 Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-ADC(3) we have an interesting result. Indeed, we have a wrong ordering in our first excited state, we find that the triplet state $1 ~^3 A_{2g}$ is lower in energy that the singlet state $B_{1g}$ in contrary to all of the results extracted in Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}. Then adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the difference of excitation energies with SF-ADC(3) it gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1 ~^3 A_{2g}$ above the singlet state $B_{1g}$. So here we have an example where the dynamical corrections are necessary to get the right chemistry.
 
+%%% FIG 3 %%%
+\begin{figure*}
+	\includegraphics[width=0.45\linewidth]{CBD_D2h}
+	\includegraphics[width=0.45\linewidth]{CBD_D4h}
+\caption{
+	Vertical excitation energies of CBD.
+	Left: $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state (see Table \ref{tab:CBD_D2h} for the raw data).
+	Right: $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3 A_{2g}$ state (see Table \ref{tab:CBD_D4h} for the raw data).
+	All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
+	\label{fig:CBD}}
+\end{figure*}
+%%% %%% %%% 
+
 %%% TABLE ?? %%%
 \begin{table}
 	\caption{
-	Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ singlet  ground state.
-	All the spin-flip calculations have been performed with a UHF reference.
+	Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
+	All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
 	\label{tab:CBD_D2h}}
 	\begin{ruledtabular}
 		\begin{tabular}{lccc}
@@ -828,8 +841,8 @@ Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-A
 %%% TABLE ?? %%%
 \begin{table}
 	\caption{
-	Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $\text{X}\,{}^1B_{1g}$ singlet ground state.
-	All the spin-flip calculations have been performed with a UHF reference.
+	Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
+	All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
 	\label{tab:CBD_D4h}}
 	\begin{ruledtabular}
 		\begin{tabular}{lccc}