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basis effects and TDA

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Pierre-Francois Loos 2020-05-20 17:17:34 +02:00
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BSEdyn.tex
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@ -69,6 +69,7 @@
\newcommand{\GW}{GW} \newcommand{\GW}{GW}
\newcommand{\stat}{\text{stat}} \newcommand{\stat}{\text{stat}}
\newcommand{\dyn}{\text{dyn}} \newcommand{\dyn}{\text{dyn}}
\newcommand{\TDA}{\text{TDA}}
% energies % energies
\newcommand{\Enuc}{E^\text{nuc}} \newcommand{\Enuc}{E^\text{nuc}}
@ -586,12 +587,12 @@ From a practical point of view, if one enforces the TDA, we obtain the very simp
\label{eq:Om1-TDA} \label{eq:Om1-TDA}
\Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}. \Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}.
\end{equation} \end{equation}
This correction can be renormalized by computing, at basically no extra cost, the renormalization factor This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the TDA,
\begin{equation} \begin{equation}
\label{eq:Z} \label{eq:Z}
Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}, Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}.
\end{equation} \end{equation}
which finally yields This finally yields
\begin{equation} \begin{equation}
\Om{m}{\text{dyn}} = \Om{m}{\text{stat}} + \Delta\Om{m}{\text{dyn}} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}. \Om{m}{\text{dyn}} = \Om{m}{\text{stat}} + \Delta\Om{m}{\text{dyn}} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}.
\end{equation} \end{equation}
@ -637,6 +638,38 @@ All the BSE calculations have been performed with our locally developed $GW$ sof
\label{sec:resdis} \label{sec:resdis}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%%
\begin{table*}
\caption{
Excitation energies (in eV) of \ce{N2} at the BSE@{\GOWO}@HF level for various basis sets.
\label{tab:N2}
}
\begin{ruledtabular}
\begin{tabular}{lddddddddd}
& \mc{3}{c}{aug-cc-pVDZ ($\Eg^{\GW} = 19.49$ eV)}
& \mc{3}{c}{aug-cc-pVTZ ($\Eg^{\GW} = 19.20$ eV)}
& \mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = $ eV)} \\
\cline{2-4} \cline{5-7} \cline{8-10}
State & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn,\TDA}$} & \tabc{$\Delta\Om{m}{\dyn}$}
& \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn,\TDA}$} & \tabc{$\Delta\Om{m}{\dyn}$}
& \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn,\TDA}$} & \tabc{$\Delta\Om{m}{\dyn}$} \\
\hline
$^1\Pi_g(n \ra \pis)$ & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.40 & & & \\
$^1\Sigma_u^-(\pi \ra \pis)$ & 9.95 & -0.44 & -0.44 & 10.11 & -0.45 & -0.45 & & & \\
$^1\Delta_u(\pi \ra \pis)$ & 10.57 & -0.41 & -0.40 & 10.75 & -0.42 & -0.41 & & & \\
$^1\Sigma_g^+$(R) & 13.72 & -0.04 & -0.04 & 13.60 & -0.03 & -0.03 & & & \\
$^1\Pi_u$(R) & 14.07 & -0.05 & -0.05 & 13.98 & -0.04 & -0.04 & & & \\
$^1\Sigma_u^+$(R) & 13.80 & -0.08 & -0.08 & 13.98 & -0.07 & -0.08 & & & \\
$^1\Pi_u$(R) & 14.22 & -0.04 & -0.03 & 14.24 & -0.03 & -0.03 & & & \\
$^3\Sigma_u^+(\pi \ra \pis)$ & 9.21 & -1.01 & -0.71 & 9.50 & -1.04 & -0.73 & & & \\
$^3\Pi_g(n \ra \pis)$ & 9.58 & -0.57 & -0.34 & 9.85 & -0.58 & -0.33 & & & \\
$^3\Delta_u(\pi \ra \pis)$ & 9.97 & -0.92 & -0.58 & 10.19 & -0.95 & -0.36 & & & \\
$^3\Sigma_u^-(\pi \ra \pis)$ & 10.71 & -0.81 & -0.68 & 10.89 & -0.82 & -0.30 & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{ \caption{
@ -656,8 +689,8 @@ All the BSE calculations have been performed with our locally developed $GW$ sof
\ce{H2O} & $^1B_1(n \ra 3s)$ & 13.58 & 8.09 & 8.00 & -0.09 & 1.007 & & & & & & 7.62 & 7.18 & 7.60 & 7.23 & 7.65 \\ \ce{H2O} & $^1B_1(n \ra 3s)$ & 13.58 & 8.09 & 8.00 & -0.09 & 1.007 & & & & & & 7.62 & 7.18 & 7.60 & 7.23 & 7.65 \\
& $^1A_2(n \ra 3p)$ & & 9.79 & 9.72 & -0.07 & 1.005 & & & & & & 9.41 & 8.84 & 9.36 & 8.89 & 9.43 \\ & $^1A_2(n \ra 3p)$ & & 9.79 & 9.72 & -0.07 & 1.005 & & & & & & 9.41 & 8.84 & 9.36 & 8.89 & 9.43 \\
& $^1A_1(n \ra 3s)$ & & 10.42 & 10.35 & -0.07 & 1.006 & & & & & & 9.99 & 9.52 & 9.96 & 9.58 & 10.00 \\ & $^1A_1(n \ra 3s)$ & & 10.42 & 10.35 & -0.07 & 1.006 & & & & & & 9.99 & 9.52 & 9.96 & 9.58 & 10.00 \\
\ce{N2} & $^1\Pi_g(n \ra \pis)$ & 19.20 & 10.11 & 9.66 & -0.45 & 1.029 & & & & & & 9.66 & 9.48 & 9.41 & 9.44 & 9.34 \\ \ce{N2} & $^1\Pi_g(n \ra \pis)$ & 19.20 & 10.42 & 9.99 & -0.42 & 1.031 & & & & & & 9.66 & 9.48 & 9.41 & 9.44 & 9.34 \\
& $^1\Sigma_u^-(\pi \ra \pis)$ & & 10.42 & 9.99 & -0.42 & 1.031 & & & & & & 10.31 & 10.26 & 10.00 & 10.32 & 9.88 \\ & $^1\Sigma_u^-(\pi \ra \pis)$ & & 10.11 & 9.66 & -0.45 & 1.029 & & & & & & 10.31 & 10.26 & 10.00 & 10.32 & 9.88 \\
& $^1\Delta_u(\pi \ra \pis)$ & & 10.75 & 10.33 & -0.42 & 1.030 & & & & & & 10.85 & 10.79 & 10.44 & 10.86 & 10.29 \\ & $^1\Delta_u(\pi \ra \pis)$ & & 10.75 & 10.33 & -0.42 & 1.030 & & & & & & 10.85 & 10.79 & 10.44 & 10.86 & 10.29 \\
& $^1\Sigma_g^+$(R) & & 13.60 & 13.57 & -0.03 & 1.003 & & & & & & 13.67 & 12.99 & 13.15 & 12.83 & 13.01 \\ & $^1\Sigma_g^+$(R) & & 13.60 & 13.57 & -0.03 & 1.003 & & & & & & 13.67 & 12.99 & 13.15 & 12.83 & 13.01 \\
& $^1\Pi_u$(R) & & 13.98 & 13.94 & -0.04 & 1.004 & & & & & & 13.64 & 13.32 & 13.43 & 13.15 & 13.22 \\ & $^1\Pi_u$(R) & & 13.98 & 13.94 & -0.04 & 1.004 & & & & & & 13.64 & 13.32 & 13.43 & 13.15 & 13.22 \\
@ -726,6 +759,25 @@ All the BSE calculations have been performed with our locally developed $GW$ sof
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}
%%% TABLE III %%%
\begin{table}
\caption{
Excitation energies (in eV) of CN3 obtained with the aug-cc-pVDZ basis set at various levels of theory.
%$\Eg^{\GW} = 13.79$ eV.
\label{tab:CN3}
}
\begin{ruledtabular}
\begin{tabular}{lcc}
& \mc{2}{c}{Excitation} \\
Method & $^1B_2(\pi \ra \pis)$ & $^3B_2(\pi \ra \pis)$ \\
\hline
BSE@{\GOWO}@HF & 7.66 & 6.52 \\
dBSE(TDA)@{\GOWO}@HF & 7.51 & 6.11 \\
FCI & 7.14 & 5.47 \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:conclusion} \label{sec:conclusion}
@ -735,7 +787,7 @@ This is the conclusion
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{ \acknowledgements{
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support. PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005. This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}} This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}