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13
BSEdyn.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-07-22 10:40:20 +0200
%% Created for Pierre-Francois Loos at 2020-07-22 22:53:55 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2020g,
Author = {P. F. Loos and D. Jacquemin},
Date-Added = {2020-07-22 22:42:52 +0200},
Date-Modified = {2020-07-22 22:43:56 +0200},
Doi = {10.1021/acs.jpclett.9b03652},
Journal = {J. Phys. Chem. Lett.},
Pages = {974},
Title = {Is ADC(3) as Accurate as CC3 for Valence and Rydberg Excitation Energies?},
Volume = {11},
Year = {2020}}
@article{Loos_2020f,
Author = {P. F. Loos and J. Authier},
Date-Added = {2020-07-22 10:38:59 +0200},

65
BSEdyn.tex
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@ -748,8 +748,10 @@ In the present study, the zeroth-order Hamiltonian [see Eq.~\eqref{eq:LR-PT}] is
The dynamical correction, however, is computed in the dTDA throughout.
As one-electron basis sets, we employ the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
It is important to mention that the small molecular systems considered here are particularly challenging for the BSE formalism which is known to work best for larger systems where the amount of screening is more important. \cite{Jacquemin_2017b,Rangel_2017}
For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted.
Various statistical quantities are reported in the following: the mean signed error (MSE), mean absolute error (MAE), root-mean-square error (RMSE), and the maximum positive [Max($+$)] and maximum negative [Max($-$)] errors.
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}, where the present perturbative correction has been implemented.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -916,9 +918,9 @@ This is quite a nice feature as it means that one does not need to compute the d
& 0.12 & 0.00 & 0.13 & 0.00 & 0.00 \\
RMSE & & & & 0.71 & 0.58 & &
& 0.54 & 0.34 & 0.19 & 0.33 & 0.00 \\
Max(+) & & & & 1.08 & 0.91 & &
Max($+$) & & & & 1.08 & 0.91 & &
& 1.06 & 0.54 & 0.44 & 0.57 & 0.00 \\
Max(-) & & & & 0.20 & -0.22 & &
Max($-$) & & & & 0.20 & -0.22 & &
& -1.77 & -0.76 & -0.02 & -0.71 & 0.00 \\
\end{tabular}
\end{ruledtabular}
@ -996,11 +998,25 @@ This is quite a nice feature as it means that one does not need to compute the d
& 0.25 & 0.21 & 0.09 & 0.22 & 0.00 \\
MSE & & & & 0.39 & 0.01 & &
& 0.21 & 0.08 & 0.04 & 0.12 & 0.00 \\
RMSE & & & & 0.44 & 0.35 & &
& 0.30 & 0.27 & 0.13 & 0.29 & 0.00 \\
Max($+$) & & & & 0.70 & 0.60 & &
& 0.63 & 0.57 & 0.29 & 0.63 & 0.00 \\
Max($-$) & & & & -0.06 & -0.74 & &
& -0.40 & -0.67 & -0.12 & -0.62 & 0.00 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set.
For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CCSD, and CC2] are also reported.
The highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference, and statistical quantities [MAE, MSE, RMSE, Max($+$), and Max($-$)] are computed with respect to these references.
For each excitation, we report the static and dynamic excitation energies, $\Om{S}{\stat}$ and $\Om{S}{\dyn}$, as well as the value of the renormalization factor $Z_S$ defined in Eq.~\eqref{eq:Z}.
As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_S$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result.
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.
A clear general trend is the consistent red shift of the static BSE excitation energies induced by the dynamical correction, as anticipated in Sec.~\ref{sec:dynW}.
%%% FIG I %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig1a}
@ -1011,21 +1027,14 @@ This is quite a nice feature as it means that one does not need to compute the d
\label{fig:SiTr-SmallMol}}
\end{figure*}
%%% FIG II %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig2}
\caption{Error (in eV) with respect to CC3 for singlet and triplet excitation energies of various molecules obtained with the aug-cc-pVDZ basis set computed within the static (white) and dynamic (colored) BSE formalism.
R stands for Rydberg state.
See Table \ref{tab:BigMol} for raw data.
\label{fig:SiTr-BigMol}}
\end{figure*}
The results gathered in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} are depicted in Fig.~\ref{fig:SiTr-SmallMol} where we report the error (with respect to the TBEs) for the singlet and triplet excitation energies computed within the static and dynamic BSE formalism.
From this figure, it is quite clear that the dynamical correction systematically improves upon its static analog, except in a very few cases.
In particular, the MAE is reduced from $0.65$ to $0.50$ eV for singlets, and from $0.39$ to $0.29$ eV for triplets.
The MSE and RMSE are also systematically improved when one takes into account dynamical effects.
The second important observation extracted from Fig.~\ref{fig:SiTr-SmallMol} is that the (singlet and triplet) Rydberg states are rather unaltered by the dynamical effects with a correction of few hundredths of eV in most cases.
The magnitude of the dynamical correction for $n \ra \pis$ and $\pi \ra \pis$ transitions is much more important: $0.3$--$0.5$ eV for singlets and $0.3$--$0.7$ eV for triplets.
Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set.
For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CCSD, and CC2] are also reported.
Finally, the highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference.
For each excitation, we report the static and dynamic excitation energies, $\Om{S}{\stat}$ and $\Om{S}{\dyn}$, as well as the value of the renormalization factor $Z_S$ defined in Eq.~\eqref{eq:Z}.
As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_S$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result.
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.
\titou{Comparison with second-order methods comes here.}
%%% TABLE III %%%
\begin{squeezetable}
@ -1070,13 +1079,31 @@ Moreover, we have observed that an iterative, self-consistent resolution [where
\\
streptocyanine & $^1B_2(\pi \ra \pis)$ & Val. & 13.79 & 7.66 & 7.51 & -0.15 & 1.019 & 7.14 \\
\hline
MAE & & & & 0.32 & 0.30 & & & 0.00 \\
MSE & & & & 0.23 & 0.00 & & & 0.00 \\
MAE & & & & 0.32 & 0.23 & & & 0.00 \\
MSE & & & & 0.30 & 0.00 & & & 0.00 \\
RMSE & & & & 0.38 & 0.29 & & & 0.00 \\
Max($+$) & & & & 0.85 & 0.54 & & & 0.00 \\
Max($-$) & & & & -0.19 & -0.73 & & & 0.00 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\end{squeezetable}
%%% FIG II %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig2}
\caption{Error (in eV) with respect to CC3 for singlet and triplet excitation energies of various molecules obtained with the aug-cc-pVDZ basis set computed within the static (white) and dynamic (colored) BSE formalism.
R stands for Rydberg state.
See Table \ref{tab:BigMol} for raw data.
\label{fig:SiTr-BigMol}}
\end{figure*}
Table \ref{tab:BigMol} reports singlet and triplet excitation energies for larger molecules at the static and dynamic BSE levels with the aug-cc-pVDZ basis set.
We also report the CC3 excitation energies computed in Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} with the same basis set.
These will be our reference as they are known to be extremely accurate ($0.03$--$0.04$ eV from the TBEs). \cite{Loos_2020g}
Error (in eV) with respect to CC3 are represented in Fig.~\ref{fig:SiTr-BigMol}.
As expected the static BSE excitation energies are much more accurate for these larger molecules with a MAE of $0.32$ eV and a MSE of $0.30$ eV.
Here again, the dynamical correction improves the accuracy of BSE by lowering the MAE and MSE to $0.23$ and $0.00$ eV, respectively.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
@ -1101,7 +1128,7 @@ This study has been (partially) supported through the EUR grant NanoX No.~ANR-17
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability}
%%%%%%%%%%%%%%%%%%%%%%%%
The data that support the findings of this study are available within the article and its {\SI}.
The data that support the findings of this study are available within the article.% and its {\SI}.
\appendix