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Pierre-Francois Loos 2020-06-08 20:54:51 +02:00
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BSEdyn.tex
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@ -733,6 +733,28 @@ All the static and dynamic BSE calculations have been performed with the softwar
}
\begin{ruledtabular}
\begin{tabular}{llddddddddd}
& & \mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}
& \mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
& \mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)} \\
\cline{3-5} \cline{6-8} \cline{9-11}
State & Nature & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
\hline
$^1\Pi_g(n \ra \pis)$ & Val. & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
$^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
$^1\Delta_u(\pi \ra \pis)$ & Val. & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
$^1\Sigma_g^+$(R) & Ryd. & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
$^1\Pi_u$(R) & Ryd. & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
$^1\Sigma_u^+$(R) & Ryd. & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
$^1\Pi_u$(R) & Ryd. & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
\\
$^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 8.69 & -0.80 & -0.72 & 8.91 & -0.97 & -0.53 & 9.06 & -1.01 & -0.80 \\
$^3\Pi_g(n \ra \pis)$ & Val. & 9.09 & -0.41 & -0.29 & 9.31 & -0.54 & -0.14 & 9.43 & -0.57 & -0.34 \\
$^3\Delta_u(\pi \ra \pis)$ & Val. & 9.49 & -0.73 & -0.62 & 9.62 & -0.89 & -0.59 & 9.74 & -0.93 & -0.99 \\
$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 10.29 & -0.65 & -0.54 & 10.34 & -0.79 & -0.43 & 10.45 & -0.82 & -0.51 \\
\hline
\\
& & \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 ($\Eg^{\GW} = 19.00$ eV)} \\
@ -755,44 +777,10 @@ All the static and dynamic BSE calculations have been performed with the softwar
$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 10.71 & -0.81 & -0.68 & 10.89 & -0.82 & -0.30 & 11.00 & -0.83 & -0.53 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Excitation energy larger than the fundamental gap.}
\end{table*}
\end{squeezetable}
%%%% TABLE I %%%
%\begin{squeezetable}
%\begin{table*}
% \caption{
% Singlet and triplet excitation energies (in eV) of \ce{N2} computed at the BSE@{\GOWO}@HF level for various basis sets.
% \label{tab:N2}
% }
% \begin{ruledtabular}
% \begin{tabular}{lddddddddd}
% & \mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}
% & \mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
% & \mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)} \\
% \cline{2-4} \cline{5-7} \cline{8-10}
% State & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
% \hline
% $^1\Pi_g(n \ra \pis)$ & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
% $^1\Sigma_u^-(\pi \ra \pis)$ & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
% $^1\Delta_u(\pi \ra \pis)$ & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
% $^1\Sigma_g^+$(R) & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
% $^1\Pi_u$(R) & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
% $^1\Sigma_u^+$(R) & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
% $^1\Pi_u$(R) & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
% \\
% $^3\Sigma_u^+(\pi \ra \pis)$ & 8.69 & -0.80 & -0.72 & 8.91 & -0.97 & -0.53 & 9.06 & -1.01 & -0.80 \\
% $^3\Pi_g(n \ra \pis)$ & 9.09 & -0.41 & -0.29 & 9.31 & -0.54 & -0.14 & 9.43 & -0.57 & -0.34 \\
% $^3\Delta_u(\pi \ra \pis)$ & 9.49 & -0.73 & -0.62 & 9.62 & -0.89 & -0.59 & 9.74 & -0.93 & -0.99 \\
% $^3\Sigma_u^-(\pi \ra \pis)$ & 10.29 & -0.65 & -0.54 & 10.34 & -0.79 & -0.43 & 10.45 & -0.82 & -0.51 \\
% \end{tabular}
% \end{ruledtabular}
% \fnt[1]{Excitation energy larger than the fundamental gap.}
%\end{table*}
%\end{squeezetable}
First, we investigate the basis set dependency of the dynamical correction as well as the validity of the dTDA (which corresponds to neglecting the dynamical correction originating from the anti-resonant part of the BSE Hamiltonian).
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 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 each corrected transition is well below this gap.
@ -905,7 +893,7 @@ In accordance with the success of the dTDA, the remaining calculations of the pr
\begin{ruledtabular}
\begin{tabular}{llldddddddddd}
& & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\%& \mc{5}{c}{Density-based methods} \\
\cline{4-8} \cline{8-13} %\cline{13-17}
\cline{4-8} \cline{9-13} %\cline{13-17}
Mol. & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$}
& \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{TBE} \\
% & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\
@ -974,8 +962,8 @@ As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of
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 %%%
%\begin{squeezetable}
\begin{table*}
\begin{squeezetable}
\begin{table}
\caption{
Singlet 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.
@ -985,52 +973,29 @@ Moreover, we have observed that an iterative, self-consistent resolution [where
\begin{tabular}{llldddddd}
& & & \mc{5}{c}{BSE@{\GOWO}@HF} \\
\cline{4-8}
Mol. & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$} & \tabc{TBE} \\
Molecule & State & Nature & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{s}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$} & \tabc{CC3} \\
\hline
acrolein & $^1A''(n \ra \pis)$ & Val. & \\
acrolein & $^1A''(n \ra \pis)$ & Val. & 11.67 & 4.62 & 4.28 & -0.35 & 1.030 & 3.77 \\
& $^1A'(n \ra \pis)$ & Val. & & 6.86 & 6.70 & -0.16 & 1.023 & 6.67 \\
& $^1A''(n \ra \pis)$ & Val. & & 7.85 & 7.71 & -0.14 & 1.012 & 6.75 \\
& $^1A'(n \ra 3s)$ & Ryd. & & 7.57 & 7.53 & -0.04 & 1.004 & 6.99 \\
\\
butadiene & $^1B_u(\pi \ra \pis)$ & Val. & 9.88 & 6.25 & 6.13 & -0.12 & 1.019 \\
& $^1A_g(\pi \ra \pis)$ & Val. & & 6.88 & 6.86 & -0.03 & 1.003 \\
& $^3B_u(\pi \ra \pis)$ & Val. & & 5.09 & 4.61 & -0.48 & 1.054 \\
butadiene & $^1B_u(\pi \ra \pis)$ & Val. & 9.88 & 6.25 & 6.13 & -0.12 & 1.019 & 6.25 \\
& $^1A_g(\pi \ra \pis)$ & Val. & & 6.88 & 6.86 & -0.03 & 1.003 & 6.68 \\
\\
diacetylene & $^1\Sigma_u^-(\pi \ra \pis)$ & Val. \\
& $^1\Delta_u(\pi \ra \pis)$ & Val. \\
& $^3\Sigma_u^+(\pi \ra \pis)$ & Val. \\
& $^3\Delta_u(\pi \ra \pis)$ & Val. \\
diacetylene & $^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 11.01 & 5.62 & 5.35 & -0.28 & 1.025 & 5.44 \\
& $^1\Delta_u(\pi \ra \pis)$ & Val. & & 5.87 & 5.63 & -0.25 & 1.024 & 5.69 \\
\\
glyoxal & $^1A_u(n \ra \pis)$ & Val. & 10.90 & 3.46 & 3.14 & -0.33 & 1.028 \\
& $^1B_g(n \ra \pis)$ & Val. & & 4.96 & 4.55 & -0.41 & 1.034 \\
& $^1B_g(n \ra \pis)$ & Val. & & & & & \\
& $^1B_u(n \ra 3p)$ & Ryd. & & & & & \\
& $^3A_u(n \ra \pis)$ & Val. & & 3.94 & 3.57 & -0.37 & 1.045 \\
& $^3B_g(n \ra \pis)$ & Val. & & 5.70 & 5.30 & -0.40 & 1.051 \\
& $^3B_u(\pi \ra \pis)$ & Val. & & 6.69 & 6.07 & -0.62 & 1.057 \\
glyoxal & $^1A_u(n \ra \pis)$ & Val. & 10.90 & 3.46 & 3.14 & -0.33 & 1.028 & 2.90 \\
& $^1B_g(n \ra \pis)$ & Val. & & 4.96 & 4.55 & -0.41 & 1.034 & 4.30 \\
& $^1B_u(n \ra 3p)$ & Ryd. & & 7.90 & 7.86 & -0.04 & 1.004 & 7.55 \\
\\
streptocyanine & $^1B_2(\pi \ra \pis)$ & Val. & 7.66 & 7.51 & -0.15 & 1.019 & 7.13 \\
& $^3B_2(\pi \ra \pis)$ & Val. & 6.52 & 6.11 & -0.41 & 1.042 & 5.52 \\
streptocyanine & $^1B_2(\pi \ra \pis)$ & Val. & 13.79 & 7.66 & 7.51 & -0.15 & 1.019 & 7.14 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%\end{squeezetable}
\end{table}
\end{squeezetable}
%%% 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}

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