more res

BSE-PES.tex

@@ -107,6 +107,7 @@
 
 %% bold in Table
 \newcommand{\bb}[1]{\textbf{#1}}
+\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
 
 % excitation energies
 \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
@@ -431,47 +432,47 @@ However, we are currently pursuing different avenues to lower this cost by compu
 %\section{Potential energy surfaces}
 %\label{sec:PES}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{N2}, \ce{CO}, \ce{BF}, \ce{F2}, and \ce{HCl}.
-The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances are gathered in Table \ref{tab:Req}.
+In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{HCl}, \ce{N2}, \ce{CO}, \ce{BF}, and , \ce{F2}.
+The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances for various basis sets are gathered in Table \ref{tab:Req}.
 Additional graphs for other basis sets can be found in the {\SI}.
 
 %%% TABLE I %%%
 \begin{table*}
 \caption{
 Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. 
-The reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in bold for visual convenience.}
+The reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in black and red bold for visual convenience, respectively.}
 \label{tab:Req}
 	
 	\begin{ruledtabular}
 	\begin{tabular}{llcccccccc}
 				&				&	\mc{8}{c}{Molecules}	\\
 									\cline{3-10}
-	Method		&	Basis		&	\ce{H2}		&	\ce{LiH}&	\ce{LiF}&	\ce{N2}	&	\ce{CO}	&	\ce{BF}	&	\ce{F2}	&	\ce{HCl}\\
+	Method		&	Basis		&	\ce{H2}		&	\ce{LiH}&	\ce{LiF}&	\ce{HCl}&	\ce{N2}	&	\ce{CO}	&	\ce{BF}	&	\ce{F2}	\\
 	\hline
-	CC3				&	cc-pVDZ		&	1.438	&	3.043	&	3.012	&	2.114	&	2.166	&	2.444	&	2.740	&	2.435	\\
-					&	cc-pVTZ		&	1.403	& \bb{3.011}&	2.961	&	2.079	&	2.143	&	2.392	&	2.669	&	2.413	\\
-					&	cc-pVQZ		& \bb{1.402}&	3.019	&	2.963	&	2.075	&	2.136	&	2.390	&	2.663	&	2.403	\\
-	CCSD			&	cc-pVDZ		&	1.438	&	3.044	&	3.006	&	2.101	&	2.149	&	2.435	&	2.695	&	2.433	\\
-					&	cc-pVTZ		&	1.403	&	3.012	&	2.954	&	2.064	&	2.126	&	2.382	&	2.629	&	2.409	\\
-					&	cc-pVQZ		&	1.402	&	3.020	&	2.953	&	2.059	&	2.118	&	2.118	&	2.621	&	2.398	\\
-	CC2				&	cc-pVDZ		&	1.426	&	3.046	&	3.026	&	2.146	&	2.187	&	2.444	&	2.710	&	2.427	\\
-					&	cc-pVTZ		&	1.393	&	3.008	&	2.995	&	2.109	&	2.163	&	2.394	&	2.664	&	2.406	\\
-					&	cc-pVQZ		&	1.391	&	2.989	&	2.982	&	2.106	&	2.156	&	2.393	&	2.665	&	2.396	\\
-	MP2				&	cc-pVDZ		&	1.426	&	3.041	&	3.010	&	2.133	&	2.166	&	2.431	&	2.681	&	2.426	\\
-					&	cc-pVTZ		&	1.393	&	3.004	&	2.968	&	2.095	&	2.144	&	2.383	&	2.636	&	2.405	\\
-					&	cc-pVQZ		&	1.391	&	3.008	&	2.970	&	2.091	&	2.137	&	2.382	&	2.634	&	2.395	\\
-	BSE@{\GOWO}@HF	&	cc-pVDZ		&	1.437	&	3.042	&	3.000	&	2.107	&	2.153	&	2.407	&	2.700	&	>2.440	\\
-					&	cc-pVTZ		&	1.404	& \bb{3.023}&	glitch	&			&			&	<2.420	&			&	<2.410	\\
-					&	cc-pVQZ		& \bb{1.399}&			&			&			&			&			&			&			\\
-	RPA@{\GOWO}@HF	&	cc-pVDZ		&	1.426	&	3.019	&	2.994	&	2.083	&	2.144	&	2.403	&	2.691	&	2.436	\\
-					&	cc-pVTZ		&	1.388	&	3.013	&	glitch	&			&			&	<2.420	&			&	<2.410	\\
-					&	cc-pVQZ		&	1.382	&			&			&			&			&			&			&			\\
-	RPAx@HF			&	cc-pVDZ		&	1.428	&	3.040	&	2.998	&	2.077	&	2.130	&	2.417	&	NaN		&	2.424	\\
-					&	cc-pVTZ		&	1.395	&	3.003	&	<2.990	&			&			&	<2.420	&			&	<2.410	\\
-					&	cc-pVQZ		&	1.394	&			&			&			&			&			&			&			\\
-	RPA@HF			&	cc-pVDZ		&	1.431	&	3.021	&	2.999	&	2.083	&	2.134	&			&	2.623	&	2.424	\\
-					&	cc-pVTZ		&	1.388	&	2.978	&	<2.990	&			&			&	2.416	&			&	<2.410	\\
-					&	cc-pVQZ		&	1.386	&			&			&			&			&	<2.420	&			&			\\
+	CC3				&	cc-pVDZ		&	1.438	&	3.043	&	3.012	&	2.435	&	2.114	&	2.166	&	2.444	&	2.740	\\
+					&	cc-pVTZ		&	1.403	& 	3.011	&	2.961	&	2.413	&	2.079	&	2.143	&	2.392	&	2.669	\\
+					&	cc-pVQZ		&\bb{1.402}	&\bb{3.019}	&	2.963	&	2.403	&	2.075	&	2.136	&	2.390	&	2.663	\\
+	CCSD			&	cc-pVDZ		&	1.438	&	3.044	&	3.006	&	2.433	&	2.101	&	2.149	&	2.435	&	2.695	\\
+					&	cc-pVTZ		&	1.403	&	3.012	&	2.954	&	2.409	&	2.064	&	2.126	&	2.382	&	2.629	\\
+					&	cc-pVQZ		&	1.402	&	3.020	&	2.953	&	2.398	&	2.059	&	2.118	&	2.118	&	2.621	\\
+	CC2				&	cc-pVDZ		&	1.426	&	3.046	&	3.026	&	2.427	&	2.146	&	2.187	&	2.444	&	2.710	\\
+					&	cc-pVTZ		&	1.393	&	3.008	&	2.995	&	2.406	&	2.109	&	2.163	&	2.394	&	2.664	\\
+					&	cc-pVQZ		&	1.391	&	2.989	&	2.982	&	2.396	&	2.106	&	2.156	&	2.393	&	2.665	\\
+	MP2				&	cc-pVDZ		&	1.426	&	3.041	&	3.010	&	2.426	&	2.133	&	2.166	&	2.431	&	2.681	\\
+					&	cc-pVTZ		&	1.393	&	3.004	&	2.968	&	2.405	&	2.095	&	2.144	&	2.383	&	2.636	\\
+					&	cc-pVQZ		&	1.391	&	3.008	&	2.970	&	2.395	&	2.091	&	2.137	&	2.382	&	2.634	\\
+	BSE@{\GOWO}@HF	&	cc-pVDZ		&	1.437	&	3.042	&	3.000	&	2.454	&	2.107	&	2.153	&	2.407	&	2.700	\\
+					&	cc-pVTZ		&	1.404	& 	3.023	&	glitch	&	<2.410	&	2.068	&			&	<2.420	&			\\
+					&	cc-pVQZ		&\rb{1.399}	&\rb{3.017}	&			&			&			&			&			&			\\
+	RPA@{\GOWO}@HF	&	cc-pVDZ		&	1.426	&	3.019	&	2.994	&	2.436	&	2.083	&	2.144	&	2.403	&	2.691	\\
+					&	cc-pVTZ		&	1.388	&	3.013	&	glitch	&	<2.410	&	2.065	&			&	<2.420	&			\\
+					&	cc-pVQZ		&	1.382	&	3.013	&			&			&			&			&			&			\\
+	RPAx@HF			&	cc-pVDZ		&	1.428	&	3.040	&	2.998	&	2.424	&	2.077	&	2.130	&	2.417	&	NaN		\\
+					&	cc-pVTZ		&	1.395	&	3.003	&	<2.990	&	<2.410	&	<2.060	&			&	<2.420	&			\\
+					&	cc-pVQZ		&	1.394	&	3.011	&			&			&			&			&			&			\\
+	RPA@HF			&	cc-pVDZ		&	1.431	&	3.021	&	2.999	&	2.424	&	2.083	&	2.134	&			&	2.623	\\
+					&	cc-pVTZ		&	1.388	&	2.978	&	<2.990	&	<2.410	&	<2.060	&			&	2.416	&			\\
+					&	cc-pVQZ		&	1.386	&	2.994	&			&			&			&			&	<2.420	&			\\
 % FROZEN CORE VERSION
 %	Method			&	Basis		&	\ce{H2}	&	\ce{LiH}&	\ce{LiF}&	\ce{N2}	&	\ce{CO}	&	\ce{BF}	&	\ce{F2}	&	\ce{HCl}\\
 %	\hline
@@ -504,15 +505,16 @@ The reference CC3 and corresponding BSE@$G_0W_0$@HF data are highlighted in bold
 	\end{ruledtabular}
 \end{table*}
 
-Let us first start with the two smallest molecules, \ce{H2} and \ce{LiH} which are both linked by covalent bonds (see Fig.~\ref{fig:PES-H2-LiH}).
-For \ce{H2}, we take as reference the full configuration interaction (FCI) energies and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
-RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly overestimate (in absolute value) the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly underestimate and overestimate the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
+Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds (see Fig.~\ref{fig:PES-H2-LiH}).
+For \ce{H2}, we take as reference the full configuration interaction (FCI) energies \cite{QP2} and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
+RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over and undershoot the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
 Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
 For example, with the cc-pVQZ basis set, BSE@{\GOWO}@HF is only off by $0.003$ bohr compared to FCI, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
 The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
 
 The scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
 In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, and they are almost perfectly parallel.
+Here again, the BSE@{\GOWO}@HF equilibrium bond length is extremely accurate ($3.017$ bohr) as compared to FCI ($3.019$ bohr).
 
 %%% FIG 1 %%%
 \begin{figure*}
@@ -534,8 +536,8 @@ However, in the case of \ce{LiF}, the attentive reader would have observed a sma
 
 %%% FIG 2 %%%
 \begin{figure*}
-	\includegraphics[width=0.49\linewidth]{LiF_GS_VTZ}
-	\includegraphics[width=0.49\linewidth]{HCl_GS_VTZ}
+	\includegraphics[height=0.35\linewidth]{LiF_GS_VTZ}
+	\includegraphics[height=0.35\linewidth]{HCl_GS_VTZ}
 \caption{
 Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
@@ -550,9 +552,9 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding as shown i
 
 %%% FIG 3 %%%
 \begin{figure*}
-	\includegraphics[width=0.33\linewidth]{N2_GS_VTZ}
-	\includegraphics[width=0.33\linewidth]{CO_GS_VTZ}
-	\includegraphics[width=0.33\linewidth]{BF_GS_VTZ}
+	\includegraphics[height=0.26\linewidth]{N2_GS_VTZ}
+	\includegraphics[height=0.26\linewidth]{CO_GS_VTZ}
+	\includegraphics[height=0.26\linewidth]{BF_GS_VTZ}
 \caption{
 Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.