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63
Manuscript/CBD.bib
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@ -1,13 +1,74 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-03-30 22:18:22 +0200 %% Created for Pierre-Francois Loos at 2022-04-01 08:15:06 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Giner_2020,
author = {Giner,Emmanuel and Scemama,Anthony and Loos,Pierre-Fran{\c c}ois and Toulouse,Julien},
date-added = {2022-04-01 08:14:54 +0200},
date-modified = {2022-04-01 08:15:06 +0200},
doi = {10.1063/5.0002892},
journal = {J. Chem. Phys.},
number = {17},
pages = {174104},
title = {A basis-set error correction based on density-functional theory for strongly correlated molecular systems},
volume = {152},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1063/5.0002892}}
@article{Giner_2018,
author = {Emmanuel Giner and Barth\'elemy Pradines and Anthony Fert\'e and Roland Assaraf and Andreas Savin and Julien Toulouse},
date-added = {2022-04-01 08:09:43 +0200},
date-modified = {2022-04-01 08:09:43 +0200},
journal = {J. Chem. Phys.},
pages = {194301},
title = {Curing Basis-Set Convergence Of Wave-Function Theory Using Density-Functional Theory: A Systematically Improvable Approach},
volume = {149},
year = {2018}}
@article{Loos_2019d,
author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
date-added = {2022-04-01 08:09:22 +0200},
date-modified = {2022-04-01 08:09:22 +0200},
doi = {10.1021/acs.jpclett.9b01176},
journal = {J. Phys. Chem. Lett.},
pages = {2931--2937},
title = {A Density-Based Basis-Set Correction for Wave Function Theory},
volume = {10},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b01103}}
@article{Loos_2021c,
author = {Loos, Pierre-Fran{\c c}ois and Comin, Massimiliano and Blase, Xavier and Jacquemin, Denis},
date-added = {2022-03-31 11:55:40 +0200},
date-modified = {2022-03-31 11:55:40 +0200},
doi = {10.1021/acs.jctc.1c00226},
journal = {J. Chem. Theory Comput.},
number = {6},
pages = {3666-3686},
title = {Reference Energies for Intramolecular Charge-Transfer Excitations},
volume = {17},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00226}}
@article{Loos_2020d,
author = {Loos, Pierre-Francois and Jacquemin, Denis},
date-added = {2022-03-31 11:55:14 +0200},
date-modified = {2022-03-31 11:55:14 +0200},
doi = {10.1021/acs.jpclett.9b03652},
journal = {J. Phys. Chem. Lett.},
number = {3},
pages = {974--980},
title = {Is ADC(3) as Accurate as CC3 for Valence and Rydberg Transition Energies?},
volume = {11},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.9b03652}}
@article{BenAmor_2020, @article{BenAmor_2020,
author = {Ben Amor,Nadia and No{\^u}s,Camille and Trinquier,Georges and Malrieu,Jean-Paul}, author = {Ben Amor,Nadia and No{\^u}s,Camille and Trinquier,Georges and Malrieu,Jean-Paul},
date-added = {2022-03-30 22:17:29 +0200}, date-added = {2022-03-30 22:17:29 +0200},

102
Manuscript/CBD.tex
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@ -32,7 +32,7 @@
\newcommand{\fnm}{\footnotemark} \newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext} \newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
%\newcommand{\SI}{\textcolor{blue}{supporting information}} \newcommand{\SupInf}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}} \newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}} \newcommand{\T}[1]{#1^{\intercal}}
@ -112,7 +112,7 @@ Therefore, one must take into account, at least, two electronic configurations t
Of course, single-reference methods are naturally unable to describe such situations. Of course, single-reference methods are naturally unable to describe such situations.
The singlet ground state, {\sBoneg}, of the square arrangement is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}), while the lowest triplet state, {\Atwog}, is a minimum on the triplet potential energy surface. The singlet ground state, {\sBoneg}, of the square arrangement is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}), while the lowest triplet state, {\Atwog}, is a minimum on the triplet potential energy surface.
Thus, the autoisomerization barrier (AB) is defined as the difference between the square and rectangular ground-state energies. Thus, the autoisomerization barrier (AB) is defined as the difference between the square and rectangular ground-state energies.
The energy of this barrier is estimated, experimentally, in the range of 1.6-10 \kcalmol, \cite{Whitman_1982} while previous multi-reference calculations yield an energy barrier in the range of 6-7 \kcalmol. \cite{Eckert-Maksic_2006} The energy of this barrier is estimated, experimentally, in the range of \SIrange{1.6}{10}{\kcalmol}, \cite{Whitman_1982} while previous multi-reference calculations yield an energy barrier in the range of \SIrange{6}{7}{\kcalmol}. \cite{Eckert-Maksic_2006}
The lowest-energy excited states of CBD in both symmetries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and the {\sBoneg} and {\Atwog} states for the square one. The lowest-energy excited states of CBD in both symmetries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and the {\sBoneg} and {\Atwog} states for the square one.
Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown. Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown.
@ -219,14 +219,13 @@ Likewise, excitation energies with respect to the singlet ground state are compu
Nowadays, spin-flip techniques are broadly accessible thanks to intensive developments in the electronic structure community (see Ref.~\onlinecite{Casanova_2020} and references therein). Nowadays, spin-flip techniques are broadly accessible thanks to intensive developments in the electronic structure community (see Ref.~\onlinecite{Casanova_2020} and references therein).
Here, we explore the spin-flip version \cite{Lefrancois_2015} of the algebraic-diagrammatic construction \cite{Schirmer_1982} (ADC) using the standard and extended second-order ADC schemes, SF-ADC(2)-s \cite{Trofimov_1997,Dreuw_2015} and SF-ADC(2)-x, \cite{Dreuw_2015} as well as its third-order version, SF-ADC(3). \cite{Dreuw_2015,Trofimov_2002,Harbach_2014} Here, we explore the spin-flip version \cite{Lefrancois_2015} of the algebraic-diagrammatic construction \cite{Schirmer_1982} (ADC) using the standard and extended second-order ADC schemes, SF-ADC(2)-s \cite{Trofimov_1997,Dreuw_2015} and SF-ADC(2)-x, \cite{Dreuw_2015} as well as its third-order version, SF-ADC(3). \cite{Dreuw_2015,Trofimov_2002,Harbach_2014}
These calculations are performed using Q-CHEM 5.2.1. \cite{qchem} These calculations are performed using Q-CHEM 5.2.1. \cite{qchem}
\titou{ADC(2.5)?} The spin-flip version of our recently proposed composite approach, namely SF-ADC(2.5), where one simply averages the SF-ADC(2)-s and SF-ADC(3) energies, is also tested in the following. \cite{Loos_2020d}
We have also carried out spin-flip calculations within the TD-DFT framework (SF-TD-DFT), \cite{Shao_2015} and these are also performed with Q-CHEM 5.2.1. \cite{qchem} We have also carried out spin-flip calculations within the TD-DFT framework (SF-TD-DFT), \cite{Shao_2015} and these are also performed with Q-CHEM 5.2.1. \cite{qchem}
The B3LYP, \cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid functionals are considered, which contain 20\%, 25\%, 50\% of exact exchange, respectively. The B3LYP, \cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid functionals are considered, which contain 20\%, 25\%, 50\% of exact exchange, respectively.
These calculations are labeled as SF-TD-BLYP, SF-TD-B3LYP, SF-TD-PBE0, and SF-TD-BH\&HLYP in the following. These calculations are labeled as SF-TD-BLYP, SF-TD-B3LYP, SF-TD-PBE0, and SF-TD-BH\&HLYP in the following.
Additionally, we have also computed SF-TD-DFT excitation energies using range-separated hybrid functionals: CAM-B3LYP,\cite{Yanai_2004a} LC-$\omega$PBE08, \cite{Weintraub_2009a} and $\omega$B97X-V. \cite{Mardirossian_2014} Additionally, we have also computed SF-TD-DFT excitation energies using range-separated hybrid (RSH) functionals: CAM-B3LYP (19\% of short-range exact exchange and 65\% at long range), \cite{Yanai_2004a} LC-$\omega$PBE08 (16.7\% of short-range exact exchange and 100\% at long range), \cite{Weintraub_2009a} and $\omega$B97X-V (16.7\% of short-range exact exchange and 100\% at long range). \cite{Mardirossian_2014}
The main difference between these range-separated functionals is their amount of exact exchange at long range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. Finally, the hybrid meta-GGA functional M06-2X (54\% of exact exchange) \cite{Zhao_2008} and the RSH meta-GGA functional M11 (42.8\% of short-range exact exchange and 100\% at long range) \cite{Peverati_2011} are also employed.
Finally, the hybrid meta-GGA functional M06-2X \cite{Zhao_2008} and the range-separated hybrid meta-GGA functional M11 \cite{Peverati_2011} are also employed.
Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999} Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1. %EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -282,6 +281,7 @@ Therefore, we rely on CASPT2(12,12)/aug-cc-pVTZ for both the {\Dtwo} and {\Dfour
Second, because the vertical transition energies are computed for a particular equilibrium geometry, we can afford to use different methods for the rectangular and square structures. Second, because the vertical transition energies are computed for a particular equilibrium geometry, we can afford to use different methods for the rectangular and square structures.
Hence, we rely on CC3/aug-cc-pVTZ to compute the equilibrium geometry of the {\oneAg} state in the rectangular ({\Dtwo}) arrangement and the restricted open-shell (RO) version of CCSD(T)/aug-cc-pVTZ to obtain the equilibrium geometry of the {\Atwog} state in the square ({\Dfour}) arrangement. Hence, we rely on CC3/aug-cc-pVTZ to compute the equilibrium geometry of the {\oneAg} state in the rectangular ({\Dtwo}) arrangement and the restricted open-shell (RO) version of CCSD(T)/aug-cc-pVTZ to obtain the equilibrium geometry of the {\Atwog} state in the square ({\Dfour}) arrangement.
These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig.~\ref{fig:CBD}). These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig.~\ref{fig:CBD}).
The cartesian coordinates of all these geometries are provided in the {\SupInf}.
Table \ref{tab:geometries} reports the key geometrical parameters obtained at these levels of theory as well as previous geometries computed by Manohar and Krylov at the CCSD(T)/cc-pVTZ level. Table \ref{tab:geometries} reports the key geometrical parameters obtained at these levels of theory as well as previous geometries computed by Manohar and Krylov at the CCSD(T)/cc-pVTZ level.
%%% TABLE I %%% %%% TABLE I %%%
@ -318,13 +318,12 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained at th
%================================================ %================================================
\subsection{Autoisomerization barrier} \subsection{Autoisomerization barrier}
\label{sec:auto} \label{sec:auto}
The results for the calculation of the automerization barrier energy with and without spin-flip methods are shown in Table \ref{tab:auto_standard}. Two types of methods are used with spin-flip, SF-TD-DFT with several functionals and SF-ADC with the variants SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3). First, one can see that there is large variations of the energy between the different functionals. Indeed if we look at the hybrid functionals B3LYP and BH\&HLYP we can see that there is a difference of around \SI{7}{\kcalmol} through all the bases, the difference in energy between the B3LYP and PBE0 functionals is much smaller with around \SI{1.5}{\kcalmol} through all the bases. We find a similar behavior regarding the RSH functionals, we find a difference of about \SIrange{8}{9}{\kcalmol} between the M06-2X and the CAM-B3LYP functionals for all bases. The results between the CAM-B3LYP and the $\omega$B97X-V are very close in energy with a difference of around \SIrange{0.1}{0.2}{\kcalmol}. The energy difference between the M11 and the M06-2X functionals is larger with \SIrange{0.6}{0.9}{\kcalmol} for the AVXZ bases and with \SI{1.7}{\kcalmol} for the 6-31+G(d) basis. For the SF-ADC methods the energy differences are smaller with \SIrange{1.7}{2.0}{\kcalmol} between the ADC(2)-s and the ADC(2)-x schemes, \SIrange{0.9}{1.6}{\kcalmol} between the ADC(2)-s and the ADC(3) schemes and \SIrange{0.4}{0.8}{\kcalmol} between the ADC(2)-x and the ADC(3) schemes.
Then we compare results for multi-reference methods, we can see a difference of about \SIrange{2.9}{3.2}{\kcalmol} through all the bases between the CASSCF(12,12) and the CASPT2(12,12) methods. These differences can be explained by the well known lack of dynamical correlation for the CASSCF method that is typically important for close-shell molecules which is the case for the ground states of the rectangular and square geometries of the CBD molecule. The results between CASPT2(12,12) and NEVPT2(12,12) are much closer with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all the bases. Finally the last results shown in Table \ref{tab:auto_standard} are the CC ones, for the autoisomerization barrier energy we consider the CCSD, CCSDT, CCSDTQ methods and the approximations of CCDT and of CCSDTQ, the CC3 and the CC4 methods, respectively. We can see that the CCSD values are higher than the other CC methods with an energy difference of around \SIrange{1.05}{1.24}{\kcalmol} between the CCSD and the CCSDT methods. The CCSDT and CCSDTQ autoisomerization barrier energies are closer with \SI{0.2}{\kcalmol} of energy difference. The energy difference between the CCSDT and its approximation CC3 is about \SIrange{0.7}{0.8}{\kcalmol} for all the bases whereas the energy difference between the CCSDTQ and its approximate version CC4 is \SI{0.1}{\kcalmol}.
%%% TABLE I %%% %%% TABLE I %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{Autoisomerization barrier (in \kcalmol) of CBD computed with various computational methods and basis sets.} \caption{Autoisomerization barrier (in \kcalmol) of CBD computed with various computational methods and basis sets.
The values in square parenthesis have been obtained by extrapolation via the procedure described in the corresponding footnote.}
\label{tab:auto_standard} \label{tab:auto_standard}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llrrrr} \begin{tabular}{llrrrr}
@ -346,13 +345,13 @@ SF-ADC(2.5) & $7.36$ & $7.76$ & $9.11$ & \\
SF-ADC(3) & $8.03$ & $8.54$ & $9.58$ \\ SF-ADC(3) & $8.03$ & $8.54$ & $9.58$ \\
CASSCF(12,12) & $10.19$ & $10.75$ & $11.59$ & $11.62$ \\ CASSCF(12,12) & $10.19$ & $10.75$ & $11.59$ & $11.62$ \\
CASPT2(12,12) & $7.24$ & $7.53$ & $8.51$ & $8.71$ \\ CASPT2(12,12) & $7.24$ & $7.53$ & $8.51$ & $8.71$ \\
NEVPT2(12,12) & $7.12$ & $7.33$ & $8.28$ & $8.49$ \\ \alert{PC-}NEVPT2(12,12) & $7.12$ & $7.33$ & $8.28$ & $8.49$ \\
CCSD & $8.31$ & $8.80$ & $9.88$ & $10.10$ \\ CCSD & $8.31$ & $8.80$ & $9.88$ & $10.10$ \\
CC3 & $6.59$ & $6.89$ & $7.88$ & $8.06$ \\ CC3 & $6.59$ & $6.89$ & $7.88$ & $8.06$ \\
CCSDT & $7.26$ & $7.64$ & $8.68$ &$\left[ 8.86\right]$\fnm[1] \\ CCSDT & $7.26$ & $7.64$ & $8.68$ &$\left[ 8.86\right]$\fnm[1] \\
CC4 & $7.40$ & $7.78$ & $\left[ 8.82\right]$\fnm[2] & $\left[ 9.00\right]$\fnm[3]\\ CC4 & $7.40$ & $7.78$ & $\left[ 8.82\right]$\fnm[2] & $\left[ 9.00\right]$\fnm[3]\\
CCSDTQ & $7.51$ & $\left[ 7.89\right]$\fnm[4]& $\left[ 8.93\right]$\fnm[5]& $\left[ 9.11\right]$\fnm[6]\\ CCSDTQ & $7.51$ & $\left[ 7.89\right]$\fnm[4]& $\left[ 8.93\right]$\fnm[5]& $\left[ 9.11\right]$\fnm[6]\\
CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\ \alert{CIPSI} & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Value obtained using CCSDT/aug-cc-pVTZ corrected by the difference between CC3/aug-cc-pVQZ and CC3/aug-cc-pVTZ.} \fnt[1]{Value obtained using CCSDT/aug-cc-pVTZ corrected by the difference between CC3/aug-cc-pVQZ and CC3/aug-cc-pVTZ.}
@ -366,19 +365,39 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
%%% %%% %%% %%% %%% %%% %%% %%%
%%% FIGURE II %%% %%% FIGURE II %%%
\begin{figure*} \begin{figure*}
\includegraphics[scale=0.5]{AB_AVTZ.pdf} \includegraphics[scale=0.5]{AB_AVTZ.pdf}
\caption{Autoisomerization barrier (in \kcalmol) of CBD at various levels of theory using the aug-cc-pVTZ basis.} \caption{Autoisomerization barrier (in \kcalmol) of CBD at various levels of theory using the aug-cc-pVTZ basis.}
%Purple, orange, green, blue, and black histograms are for SF-TD-DFT, SF-ADC, multi-reference methods, CC and TBE.}
\label{fig:AB} \label{fig:AB}
\end{figure*} \end{figure*}
Figure \ref{fig:AB} shows the autoisomerization barrier (AB) energies for the CBD molecule for the various used methods. We see the large variations of the AB energy with the different DFT functionals with some of them giving an energy of almost 20 \kcalmol compared to the 8.93 \kcalmol of the TBE. Nevertheless, we have that some functionals, BH\&HLYP, M06-2X or M11, give comparable results to SF-ADC or to multi-reference methods. For SF-ADC and multi-reference methods we get small energy differences compared to the TBE value. Note that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the result and even increase the energy error to the TBE value. We can also notice that, as previously stated, CASSCF provide a larger energy error compared to CASPT2 and NEVPT2 due to the lack of dynamical correlation. Finally, CC methods show also good results compared to the TBE. %We see that CCSD presents a larger error and that taking into account the triple excitations improves the result.
%%% %%% %%% %%% %%% %%% %%% %%%
The results concerning the autoisomerization barrier are reported in Table \ref{tab:auto_standard} for various basis sets and shown in Fig.~\ref{fig:AB} for the aug-cc-pVTZ basis.
First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as \SI{10}{\kcalmol} for a given basis set between the different functionals.
Nonetheless, it is clear that the performance of a given functional is directly linked to the amount of exact exchange at short range.
Indeed, hybrid functionals with a large fraction of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
However, they are still off by \SIrange{1}{3}{\kcalmol} from the TBE reference value.
For the RSH functionals, the autoisomerization barrier is much less sensitive to the amount of longe-range exact exchange.
Another important feature of SF-TD-DFT is the fast convergence of the energy barrier with the size of the basis set. \cite{Loos_2019d}
With the augmented double-$\zeta$ basis, the SF-TD-DFT results are basically converged to sub-\kcalmol accuracy, which is a drastic improvement compared to wave function approaches where this type of convergence is reached with the augmented triple-$\zeta$ basis.
For the SF-ADC family of methods, the energy differences are much smaller with a maximum deviation of \SI{2.0}{\kcalmol} between different versions.
In particular, we observe that SF-ADC(2)-s and SF-ADC(3) under- and overestimate the autoisomerization barrier, respectively, making SF-ADC(2.5) a good compromise with an error of only \SI{0.18}{\kcalmol} with the aug-cc-pVTZ basis.
We note that SF-ADC(2)-x is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3).
Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC methods.
\alert{For the smaller active space, we have...}
Concerning the multi-reference approaches with the (12e,12o) active space, we see a difference of the order of \SI{3}{\kcalmol} through all the bases between CASSCF and the second-order variants (CASPT2 and NEVPT2).
These differences can be explained by the well known lack of dynamical correlation at the CASSCF level.
The deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all the bases.
\alert{Finally the last results shown in Table \ref{tab:auto_standard} are the CC ones, for the autoisomerization barrier energy we consider the CCSD, CCSDT, CCSDTQ methods and the approximations of CCSDT and of CCSDTQ, the CC3 and the CC4 methods, respectively.
We can see that the CCSD values are higher than the other CC methods with an energy difference of around \SIrange{1.05}{1.24}{\kcalmol} between the CCSD and the CCSDT methods. The CCSDT and CCSDTQ autoisomerization barrier energies are closer with \SI{0.2}{\kcalmol} of energy difference.
The energy difference between the CCSDT and its approximation CC3 is about \SIrange{0.7}{0.8}{\kcalmol} for all the bases whereas the energy difference between the CCSDTQ and its approximate version CC4 is \SI{0.1}{\kcalmol}.}
%================================================ %================================================
%================================================ %================================================
@ -398,8 +417,8 @@ For the XMS-CASPT2(4,4) only the {\twoAg} state is described with values similar
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
Spin-flip TD-DFT vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state. Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.}
\label{tab:sf_tddft_D2h}} \label{tab:sf_tddft_D2h}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llrrr} \begin{tabular}{llrrr}
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\ & \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
@ -476,8 +495,9 @@ SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
Vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state. Vertical excitation energies (with respect to the {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.
\label{tab:D2h}} The values in square parenthesis have been obtained by extrapolation via the procedure described in the corresponding footnote.}
\label{tab:D2h}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llrrr} \begin{tabular}{llrrr}
& & \mc{3}{c}{Excitation energies (eV)} \\ & & \mc{3}{c}{Excitation energies (eV)} \\
@ -560,7 +580,8 @@ Figure \ref{fig:D2h} shows the vertical energies of the studied excited states d
\begin{figure*} \begin{figure*}
%width=0.8\linewidth %width=0.8\linewidth
\includegraphics[scale=0.5]{D2h.pdf} \includegraphics[scale=0.5]{D2h.pdf}
\caption{Vertical energies of the {\tBoneg}, {\sBoneg} and {\twoAg} states for the {\Dtwo} geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.} \caption{Vertical excitation energies of the {\tBoneg}, {\sBoneg}, and {\twoAg} states at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state using the aug-cc-pVTZ basis.}
%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
\label{fig:D2h} \label{fig:D2h}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -578,8 +599,8 @@ Again, we have computed the vertical energies at the CCSDT and CCSDTQ level as w
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
Standard vertical excitation energies (with respect to the singlet {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state. Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.}
\label{tab:sf_D4h}} \label{tab:sf_D4h}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llrrr} \begin{tabular}{llrrr}
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\ & \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
@ -646,8 +667,9 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
Standard vertical excitation energies (with respect to the singlet {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state. Vertical excitation energies (with respect to the {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.
\label{tab:D4h}} The values in square brackets have been obtained by extrapolation via the procedure described in the corresponding footnote.}
\label{tab:D4h}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llrrr} \begin{tabular}{llrrr}
& \mc{3}{r}{Excitation energies (eV)} \hspace{0.1cm}\\ & \mc{3}{r}{Excitation energies (eV)} \hspace{0.1cm}\\
@ -706,8 +728,8 @@ PC-NEVPT2(12,12) & 6-31+G(d) & $0.189$ & $1.579$ & $2.020$ \\
& aug-cc-pVDZ & $0.156$ & $1.530$ & $1.854$ \\ & aug-cc-pVDZ & $0.156$ & $1.530$ & $1.854$ \\
& aug-cc-pVTZ & $0.131$ & $1.476$ & $1.756$ \\ & aug-cc-pVTZ & $0.131$ & $1.476$ & $1.756$ \\
& aug-cc-pVQZ & $0.126$ & $1.460$ & $1.727$ \\[0.1cm] & aug-cc-pVQZ & $0.126$ & $1.460$ & $1.727$ \\[0.1cm]
CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\ CIPSI & 6-31+G(d) & $0.201\pm 0.003$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
& aug-cc-pVDZ & $0.1570\pm 0.0030$ & $1.587\pm 0.005$ & $2.102\pm 0.027$ \\ & aug-cc-pVDZ & $0.157\pm 0.003$ & $1.587\pm 0.005$ & $2.102\pm 0.027$ \\
& aug-cc-pVTZ & $0.169\pm 0.029$ & $1.63\pm 0.05$ & \\ & aug-cc-pVTZ & $0.169\pm 0.029$ & $1.63\pm 0.05$ & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
@ -727,14 +749,15 @@ Figure \ref{fig:D4h} shows the vertical energies of the studied excited states d
\begin{figure*} \begin{figure*}
%width=0.8\linewidth %width=0.8\linewidth
\includegraphics[scale=0.5]{D4h.pdf} \includegraphics[scale=0.5]{D4h.pdf}
\caption{Vertical energies of the {\Atwog}, {\Aoneg} and {\Btwog} states for the {\Dfour} geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.} \caption{Vertical excitation energies (in \si{\eV}) of the {\Atwog}, {\Aoneg}, and {\Btwog} states at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state using the aug-cc-pVTZ basis.}
%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
\label{fig:D4h} \label{fig:D4h}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%% %%% %%%
\subsubsection{Theoretical best estimates} \subsubsection{Theoretical best estimates}
\label{sec:TBE} \label{sec:TBE}
Table \ref{tab:TBE} shows the energy differences, for the autoisomerization barrier (AB) and the different states considered, between the various methods treated and the Theoretical Best Estimate (TBE) at the aug-cc-pVTZ level for the AB and the states. The percentage \% T1 shown in parentheses for the excited states of the {\Dtwo} geometry is a metric that gives the percentage of single excitation calculated at the CC3/aug-cc-pVTZ level and it allows us to characterize the transition. First, we look at the AB energy difference. SF-TD-DFT shows large variations of the energy with errors of \SIrange{1.42}{10.81}{\kcalmol} compared to the TBE value. SF-ADC schemes provide smaller errors with \SIrange{0.30}{1.44}{\kcalmol} where we have that the SF-ADC(2)-x gives a worse error than the SF-ADC(2)-s method. CC methods also give small energy differences with \SIrange{0.11}{1.05}{\kcalmol} and where the CC4 provides an energy very close to the TBE one. Table \ref{tab:TBE} shows the energy differences, for the autoisomerization barrier (AB) and the different states considered, between the various methods treated and the Theoretical Best Estimate (TBE) at the aug-cc-pVTZ level for the AB and the states. The percentage $\%T_1$ shown in parentheses for the excited states of the {\Dtwo} geometry is a metric that gives the percentage of single excitation calculated at the CC3/aug-cc-pVTZ level and it allows us to characterize the transition. First, we look at the AB energy difference. SF-TD-DFT shows large variations of the energy with errors of \SIrange{1.42}{10.81}{\kcalmol} compared to the TBE value. SF-ADC schemes provide smaller errors with \SIrange{0.30}{1.44}{\kcalmol} where we have that the SF-ADC(2)-x gives a worse error than the SF-ADC(2)-s method. CC methods also give small energy differences with \SIrange{0.11}{1.05}{\kcalmol} and where the CC4 provides an energy very close to the TBE one.
Then we look at the vertical energy errors for the {\Dtwo} structure. First we consider the {\tBoneg} state and we look at the SF-TD-DFT results. We see that increasing the amount of exact exchange in the functional give closer results to the TBE, indeed we have \SI{0.24}{\eV} and \SI{0.22}{\eV} of errors for the B3LYP and the PBE0 functionals, respectively whereas we have an error of \SI{0.08}{\eV} for the BH\&HLYP functional. For the other functionals we have errors of \SIrange{0.10}{0.43}{\eV}, note that for this state the M06-2X functional gives the same result than the TBE. We can also notice that all the functionals considered overestimate the vertical energies. The ADC schemes give closer energies with errors of \SIrange{0.04}{0.08}{\eV}, note that ADC(2)-x does not improve the result compared to ADC(2)-s and that ADC(3) understimate the vertical energy whereas ADC(2)-s and ADC(2)-x overestimate the vertical energy. The CC3 and CCSDT results provide energy errors of \SIrange{0.05}{0.06}{\eV} respectively. Then we go through the multi-reference methods with the two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. For the smaller active space we have errors of \SIrange{0.05}{0.21}{\eV}, the largest error comes from CASSCF(4,4) which is improved by CASPT2(4,4) that gives the smaller error. Then for the largest active space multi-reference methods provide energy errors of \SIrange{0.02}{0.22}{\eV} with again the largest error coming from CASSCF(12,12) which is again improved by CASPT2(12,12) gives the smaller error. Then we look at the vertical energy errors for the {\Dtwo} structure. First we consider the {\tBoneg} state and we look at the SF-TD-DFT results. We see that increasing the amount of exact exchange in the functional give closer results to the TBE, indeed we have \SI{0.24}{\eV} and \SI{0.22}{\eV} of errors for the B3LYP and the PBE0 functionals, respectively whereas we have an error of \SI{0.08}{\eV} for the BH\&HLYP functional. For the other functionals we have errors of \SIrange{0.10}{0.43}{\eV}, note that for this state the M06-2X functional gives the same result than the TBE. We can also notice that all the functionals considered overestimate the vertical energies. The ADC schemes give closer energies with errors of \SIrange{0.04}{0.08}{\eV}, note that ADC(2)-x does not improve the result compared to ADC(2)-s and that ADC(3) understimate the vertical energy whereas ADC(2)-s and ADC(2)-x overestimate the vertical energy. The CC3 and CCSDT results provide energy errors of \SIrange{0.05}{0.06}{\eV} respectively. Then we go through the multi-reference methods with the two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. For the smaller active space we have errors of \SIrange{0.05}{0.21}{\eV}, the largest error comes from CASSCF(4,4) which is improved by CASPT2(4,4) that gives the smaller error. Then for the largest active space multi-reference methods provide energy errors of \SIrange{0.02}{0.22}{\eV} with again the largest error coming from CASSCF(12,12) which is again improved by CASPT2(12,12) gives the smaller error.
@ -753,7 +776,7 @@ Finally we look at the vertical energy errors for the \Dfour structure. First, w
%%% TABLE I %%% %%% TABLE I %%%
%\begin{squeezetable} %\begin{squeezetable}
%\begin{table*} %\begin{table*}
% \caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage T1 calculated at the CC3/aug-cc-pVTZ level.} % \caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage $\%T_1$ calculated at the CC3/aug-cc-pVTZ level.}
% %
% \label{tab:TBE} % \label{tab:TBE}
% \begin{ruledtabular} % \begin{ruledtabular}
@ -807,15 +830,16 @@ Finally we look at the vertical energy errors for the \Dfour structure. First, w
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{Energy differences between the various methods and the reference TBE values. \caption{Energy differences between the various methods and the reference TBE values.
Note that AB stands for the autoisomerization barrier. Note that AB stands for the autoisomerization barrier and is reported in \si{\kcalmol}.
The numbers reported in parenthesis are the percentage of single excitations involved in the transition ($\%T_1$) calculated at the CC3/aug-cc-pVTZ level.} The numbers reported in parenthesis are the percentage of single excitations involved in the transition ($\%T_1$) calculated at the CC3/aug-cc-pVTZ level.
The values between square brackets have been obtained by extrapolation via the procedure described in the corresponding footnote.}
\label{tab:TBE} \label{tab:TBE}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrrrrrrr} \begin{tabular}{lrrrrrrr}
%\begin{tabular}{*{1}{*{8}{l}}} %\begin{tabular}{*{1}{*{8}{l}}}
& &\mc{3}{c}{{\Dtwo} excitation energies (eV)} & \mc{3}{c}{{\Dfour} excitation energies (eV)} \\ & &\mc{3}{c}{{\Dtwo} excitation energies (eV)} & \mc{3}{c}{{\Dfour} excitation energies (eV)} \\
\cline{3-5} \cline{6-8} \cline{3-5} \cline{6-8}
Method & AB (\si{\kcalmol}) & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\twoAg}(1\%) & {\Atwog} & {\Aoneg} & {\Btwog} \\ Method & AB & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\twoAg}(1\%) & {\Atwog} & {\Aoneg} & {\Btwog} \\
\hline \hline
SF-TD-B3LYP & $10.41$ & $0.241$ & $-0.926$ & $-0.161$ & $-0.164$ & $-1.028$ & $-1.501$ \\ SF-TD-B3LYP & $10.41$ & $0.241$ & $-0.926$ & $-0.161$ & $-0.164$ & $-1.028$ & $-1.501$ \\
SF-TD-PBE0 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\ SF-TD-PBE0 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\
@ -862,7 +886,6 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
\fnt[6]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-x/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.} \fnt[6]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-x/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[7]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(3)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.} \fnt[7]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(3)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[8]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.} \fnt[8]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
@ -875,19 +898,24 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
\label{sec:conclusion} \label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have considered the automerization barrier (AB) energy and the vertical energies of the cyclobutadiene (CBD) molecule in the square ({\Dfour}) and rectangular ({\Dtwo}) geometries. For the AB and vertical energies we have defined theoretical best estimates (TBEs) by using the CCSDTQ/aug-cc-pVTZ values when we were able to obtain them. Otherwise we got the CCSDTQ/aug-cc-pVTZ values by correcting the CCSDTQ/aug-cc-pVDZ values by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ (Eq.~\eqref{eq:aug-cc-pVTZ}) and we obtain the CCSDTQ/aug-cc-pVDZ values by correcting the CCSDTQ/6-31G+(d) values by the difference between CC4/aug-cc-pVDZ and CC4/6-31G+(d) (Eq.~\eqref{eq:aug-cc-pVDZ}). When the CC4/aug-cc-pVTZ values were not obtained we corrected the CC4/aug-cc-pVDZ values by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ to obtain them (Eq.~\eqref{eq:CC4_aug-cc-pVTZ}). If the CC4 values have not been obtained then we used the same scheme that we just described but by using the CCSDT values. If neither the CC4 and CCSDTQ values were not available then we used the NEVPT2(12,12)/aug-cc-pVTZ values. We have considered the automerization barrier energy and the vertical energies of the cyclobutadiene (CBD) molecule in the square ({\Dfour}) and rectangular ({\Dtwo}) geometries. For the AB and vertical energies we have defined theoretical best estimates (TBEs) by using the CCSDTQ/aug-cc-pVTZ values when we were able to obtain them. Otherwise we got the CCSDTQ/aug-cc-pVTZ values by correcting the CCSDTQ/aug-cc-pVDZ values by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ (Eq.~\eqref{eq:aug-cc-pVTZ}) and we obtain the CCSDTQ/aug-cc-pVDZ values by correcting the CCSDTQ/6-31G+(d) values by the difference between CC4/aug-cc-pVDZ and CC4/6-31G+(d) (Eq.~\eqref{eq:aug-cc-pVDZ}). When the CC4/aug-cc-pVTZ values were not obtained we corrected the CC4/aug-cc-pVDZ values by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ to obtain them (Eq.~\eqref{eq:CC4_aug-cc-pVTZ}). If the CC4 values have not been obtained then we used the same scheme that we just described but by using the CCSDT values. If neither the CC4 and CCSDTQ values were not available then we used the NEVPT2(12,12)/aug-cc-pVTZ values.
In order to provide a benchmark of the AB and vertical energies we used coupled-cluster (CC) methods with doubles (CCSD), with triples (CCSDT and CC3) and with quadruples (CCSTQ and CC4). Due to the presence of multi-configurational states we used multi-reference methods (CASSCF, CASPT2 and NEVPT2) with two active spaces ((4,4) and (12,12)). We also used spin-flip (SF-) within two frameworks, in TD-DFT with various global and range-separated hybrids functionals, and in ADC with the ADC(2)-s, ADC(2)-x and ADC(3) schemes. The CC methods provide good results for the AB and vertical energies, however in the case of multi-configurational states CC with only triples is not sufficient and we have to include the quadruples to correctly describe these states. multi-configurational methods also provide very solid results for the largest active space with second order correction (CASPT2 and NEVPT2). In order to provide a benchmark of the AB and vertical energies we used coupled-cluster (CC) methods with doubles (CCSD), with triples (CCSDT and CC3) and with quadruples (CCSTQ and CC4). Due to the presence of multi-configurational states we used multi-reference methods (CASSCF, CASPT2 and NEVPT2) with two active spaces ((4,4) and (12,12)). We also used spin-flip (SF-) within two frameworks, in TD-DFT with various global and range-separated hybrids functionals, and in ADC with the ADC(2)-s, ADC(2)-x and ADC(3) schemes. The CC methods provide good results for the AB and vertical energies, however in the case of multi-configurational states CC with only triples is not sufficient and we have to include the quadruples to correctly describe these states. multi-configurational methods also provide very solid results for the largest active space with second order correction (CASPT2 and NEVPT2).
With SF-TD-DFT the quality of the results are, of course, dependent on the functional but for the doubly excited states we have solid results. In SF-ADC we have very good results compared to the TBEs even for the doubly excited states, nevertheless the ADC(2)-x scheme give almost systematically worse results than the ADC(2)-s ones and using the ADC(3) scheme does not always provide better values. With SF-TD-DFT the quality of the results are, of course, dependent on the functional but for the doubly excited states we have solid results. In SF-ADC we have very good results compared to the TBEs even for the doubly excited states, nevertheless the ADC(2)-x scheme give almost systematically worse results than the ADC(2)-s ones and using the ADC(3) scheme does not always provide better values.
The description of the excited states of the {\Dtwo} structure give rise to good agreement between the single reference and multi-configurational methods due to the large T1 percentage of the first two excited states. When this percentage is much smaller as in the case of the doubly excited state {\twoAg} the spin-flip methods show very good results within the ADC framework but even at the TD-DFT level spin-flip display good results compared to the TBE value. As said in the discussion, for the {\Dfour} geometry, the description of excited states is harder because of the strong multi-configurational character where SF-TD-DFT can present more than 1 eV of error compared to the TBE. However, SF-ADC can show error of around \SIrange{0.1}{0.2}{\eV} which can be better than the multi-configurational methods results. The description of the excited states of the {\Dtwo} structure give rise to good agreement between the single reference and multi-configurational methods due to the large $\%T_1$ percentage of the first two excited states. When this percentage is much smaller as in the case of the doubly excited state {\twoAg} the spin-flip methods show very good results within the ADC framework but even at the TD-DFT level spin-flip display good results compared to the TBE value. As said in the discussion, for the {\Dfour} geometry, the description of excited states is harder because of the strong multi-configurational character where SF-TD-DFT can present more than 1 eV of error compared to the TBE. However, SF-ADC can show error of around \SIrange{0.1}{0.2}{\eV} which can be better than the multi-configurational methods results.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{ \acknowledgements{
EM, AS, and PFL acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).} EM, AS, and PFL acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\nocite{*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Included in the {\SupInf} are the raw data and the cartesian coordinates of the various optimized geometries.
%================================================
\bibliography{CBD} \bibliography{CBD}
\end{document} \end{document}