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Manuscript/CBD.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-04-04 11:49:40 +0200
%% Created for Pierre-Francois Loos at 2022-04-05 11:03:04 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Head-Gordon_1994,
author = {M. Head-Gordon and R. J. Rico and M. Oumi and T. J. Lee},
date-added = {2022-04-04 22:56:17 +0200},
date-modified = {2022-04-04 22:56:17 +0200},
doi = {10.1016/0009-2614(94)00070-0},
journal = {Chem. Phys. Lett.},
pages = {21--29},
title = {A Doubles Correction To Electronic Excited States From Configuration Interaction In The Space Of Single Substitutions},
volume = {219},
year = {1994},
bdsk-url-1 = {https://doi.org/10.1016/0009-2614(94)00070-0}}
@article{Head-Gordon_1995,
author = {Head-Gordon, M. and Maurice, D. and Oumi, M.},
date-added = {2022-04-04 22:56:17 +0200},
date-modified = {2022-04-04 22:56:17 +0200},
doi = {10.1016/0009-2614(95)01111-L},
journal = {Chem. Phys. Lett.},
pages = {114--121},
title = {A Perturbative Correction to Restricted Open-Shell Configuration-Interaction with Single Substitutions for Excited-States of Radicals},
volume = {246},
year = {1995},
bdsk-url-1 = {https://doi.org/10.1016/0009-2614(95)01111-L}}
@article{Giner_2019,
author = {E. Giner and A. Scemama and J. Toulouse and P. F. Loos},
date-added = {2022-04-04 11:49:35 +0200},
@ -3617,19 +3641,6 @@
year = {2005},
bdsk-url-1 = {https://dx.doi.org/10.1021/cr0505627}}
@article{Piecuch_2002,
author = {Piotr Piecuch and Karol Kowalski and Ian S. O. Pimienta and Michael J. Mcguire},
date-added = {2022-03-21 21:51:24 +0100},
date-modified = {2022-03-21 21:58:10 +0100},
doi = {10.1080/0144235021000053811},
journal = {Int. Rev. Phys. Chem.},
pages = {527-655},
publisher = {Taylor & Francis},
title = {Recent advances in electronic structure theory: Method of moments of coupled-cluster equations and renormalized coupled-cluster approaches},
volume = {21},
year = {2002},
bdsk-url-1 = {https://doi.org/10.1080/0144235021000053811}}
@article{VanderLugt_1969,
author = {{Van der Lugt}, W. Th. A. M. and Oosterhoff, Luitzen J.},
date-added = {2022-03-21 21:37:07 +0100},

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Manuscript/CBD.tex
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@ -67,7 +67,7 @@
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\CEISAM}{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
\title{Reference Energies for Cyclobutadiene: Autoisomerization and Excited States}
\title{Reference Energies for Cyclobutadiene: Automerization and Excited States}
\author{Enzo \surname{Monino}}
\email{emonino@irsamc.ups-tlse.fr}
@ -85,10 +85,10 @@
\begin{abstract}
The cyclobutadiene molecule is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods.
Indeed, due to its high spatial symmetry, especially at the $D_{4h}$ square geometry but also in the $D_{2h}$ rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multi-configurational characters and single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations.
In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the autoisomerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ equilibrium structures.
In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the automerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ equilibrium structures.
In particular, selected configuration interaction (SCI), multi-reference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed.
The spin-flip formalism, which is known to provide a correct description of states with multi-configurational character, is also tested within TD-DFT (where numerous exchange-correlation functionals are considered) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)].
A theoretical best estimate is defined for the autoisomerization barrier and for each vertical transition energy.
A theoretical best estimate is defined for the automerization barrier and for each vertical transition energy.
\end{abstract}
\maketitle
@ -99,7 +99,7 @@ A theoretical best estimate is defined for the autoisomerization barrier and for
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis or in solar cell technology, \cite{Delgado_2010} none of the currently existing methods has shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, environment effects and many other possible factors.
Indeed, each computational model has its own theoretical and/or technical limitations and the number of possible chemical scenarios is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a,Hait_2021,Zobel_2021}
Indeed, each computational model has its own theoretical and/or technical limitations and the number of possible chemical scenarios is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002b,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a,Hait_2021,Zobel_2021}
Speaking of difficult tasks, the cyclobutadiene (CBD) molecule has been a real challenge for experimental and theoretical chemistry for many decades. \cite{Bally_1980} Due to its antiaromaticity \cite{Minkin_1994} and large angular strain, \cite{Baeyer_1885} CBD presents a high reactivity which made its synthesis a particularly difficult exercise.
In the {\Dfour} symmetry, the simple H\"uckel molecular orbital theory (wrongly) predicts a triplet ground state (Hund's rule) with two singly-occupied frontier orbitals that are degenerate by symmetry, while state-of-the-art \textit{ab initio} methods (correctly) predict an open-shell singlet ground state.
@ -111,7 +111,7 @@ However, in the {\Dfour} symmetry, the {\sBoneg} ground state has two singly occ
Therefore, one must take into account, at least, two electronic configurations to properly model this multi-configurational scenario.
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.
Thus, the autoisomerization barrier (AB) is defined as the difference between the square and rectangular ground-state energies.
Thus, the automerization 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 \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.
@ -137,7 +137,7 @@ Obviously, spin-flip methods have their own flaws, especially spin contamination
One can address part of this issue by increasing the excitation order or by complementing the spin-incomplete configuration set with the missing configurations, \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
both solutions being associated with an additional computational cost.
In the present work, we investigate the accuracy of each family of computational methods mentioned above on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
In the present work, we investigate the accuracy of each family of computational methods mentioned above on the automerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
Computational details are reported in Sec.~\ref{sec:compmet}.
Section \ref{sec:res} is devoted to the discussion of our results.
Finally, our conclusions are drawn in Sec.~\ref{sec:conclusion}.
@ -147,7 +147,7 @@ Finally, our conclusions are drawn in Sec.~\ref{sec:conclusion}.
\includegraphics[width=\linewidth]{figure1}
\caption{Pictorial representation of the ground and excited states of CBD and the properties under investigation.
The singlet ground state (S) and triplet (T) properties are represented in black and red, respectively.
The autoisomerization barrier (AB) is also represented.}
The automerization barrier (AB) is also represented.}
\label{fig:CBD}
\end{figure}
%%% %%% %%% %%%
@ -272,8 +272,8 @@ Note that, due to error bar inherently linked to the CIPSI calculations (see Sub
%================================================
\subsection{Geometries}
\label{sec:geometries}
Two different sets of geometries obtained with different levels of theory are considered for the autoisomerization barrier and the excited states of the CBD molecule.
First, because the autoisomerization barrier is obtained as a difference of energies computed at distinct geometries, it is paramount to obtain these at the same level of theory.
Two different sets of geometries obtained with different levels of theory are considered for the automerization barrier and the excited states of the CBD molecule.
First, because the automerization barrier is obtained as a difference of energies computed at distinct geometries, it is paramount to obtain these at the same level of theory.
However, due to the fact that the ground state of the square arrangement is a transition state of singlet open-shell nature, it is technically difficult to optimize the geometry with high-order CC methods.
Therefore, we rely on CASPT2(12,12)/aug-cc-pVTZ for both the {\Dtwo} and {\Dfour} ground-state structures.
(Note that these optimizations are done without IPEA shift.)
@ -314,13 +314,13 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained at th
%%% %%% %%% %%%
%================================================
\subsection{Autoisomerization barrier}
\subsection{automerization barrier}
\label{sec:auto}
%%% TABLE I %%%
\begin{squeezetable}
\begin{table}
\caption{Autoisomerization barrier (in \kcalmol) of CBD computed with various computational methods and basis sets.
\caption{automerization 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}
\begin{ruledtabular}
@ -372,23 +372,23 @@ CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[ 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\
%%% FIGURE II %%%
\begin{figure*}
\includegraphics[width=\linewidth]{AB_AVTZ}
\caption{Autoisomerization barrier (in \si{\kcalmol}) of CBD at various levels of theory using the aug-cc-pVTZ basis.}
\caption{automerization barrier (in \si{\kcalmol}) of CBD at various levels of theory using the aug-cc-pVTZ basis.}
\label{fig:AB}
\end{figure*}
%%% %%% %%% %%%
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.
The results concerning the automerization 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} between the different functionals for a given basis set.
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}{4}{\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.
For the RSH functionals, the automerization 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}{\kcalmol} between different versions.
In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which scale as $\order*{N^5}$ and $\order*{N^6}$ respectively (where $N$ is the number of basis functions), under- and overestimate the autoisomerization barrier, making SF-ADC(2.5) a good compromise with an error of only \SI{0.18}{\kcalmol} with the aug-cc-pVTZ basis.
In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which scale as $\order*{N^5}$ and $\order*{N^6}$ respectively (where $N$ is the number of basis functions), under- and overestimate the automerization barrier, making SF-ADC(2.5) a good compromise with an error of only \SI{0.18}{\kcalmol} with the aug-cc-pVTZ basis.
Nonetheless, at a $\order*{N^5}$ computational scaling, SF-ADC(2)-s is particularly accurate, even compared to high-order CC methods (see below).
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.
@ -582,7 +582,7 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
\end{figure*}
%%% %%% %%% %%%
Table \ref{tab:sf_tddft_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the singlet and triplet vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
Table \ref{tab:sf_tddft_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation energies with respect to the level of theory is illustrated in Fig.~\ref{fig:D2h}.
At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in the {\tBoneg}, {\sBoneg}, and {\twoAg} are 99\%, 95\%, and 1\%, respectively.
@ -592,7 +592,7 @@ First, let us discuss basis set effects at the SF-TD-DFT level.
As expected, these are found to be small and the results are basically converged to the basis set limit with the triple-$\zeta$ basis, which is definitely not the case for the wave function methods. \cite{Giner_2019}
Regarding now the accuracy of the vertical excitation energies, again, we clearly see that, for each transition, the functionals with the largest amount of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) are the most accurate.
Functionals with large amount of exact exchange are known to perform best in the SF-TD-DFT framework as the Hartree-Fock exchange term is the only non-vanishing term in the spin-flip block. \cite{Shao_2003}
However, their overall accuracy remains average especially for the singlet states, {\sBoneg} and {\twoAg}, with error of the order of \SIrange{0.2}{0.5}{\eV} compared to the TBEs for the best functionals.
However, their overall accuracy remains average especially for the singlet states, {\sBoneg} and {\twoAg}, with error of the order of \SIrange{0.2}{0.5}{\eV} compared to the TBEs.
The triplet state, {\tBoneg}, is much better described with errors below \SI{0.1}{\eV}.
Note that, as evidenced by the data reported in {\SupInf}, none of these states exhibit a strong spin contamination.
@ -610,15 +610,15 @@ Note that, as evidenced by the data reported in {\SupInf}, none of these states
Second, we discuss the various SF-ADC schemes, \ie, SF-ADC(2)-s, SF-ADC(2)-x, and SF-ADC(3).
At the SF-ADC(2)-s level, going from the smallest 6-31+G(d) basis to the largest aug-cc-pVQZ basis, we see a small decrease in vertical excitation energies of about \SI{0.03}{\eV} for the {\tBoneg} state and around \SI{0.06}{\eV} for {\twoAg} state, while the transition energy of the {\sBoneg} state drops more significantly by about \SI{0.2}{\eV}.
(The SF-ADC(2)-x and SF-ADC(3) calculations with aug-cc-pVQZ were not feasible with our computational resources.)
[The SF-ADC(2)-x and SF-ADC(3) calculations with aug-cc-pVQZ were not feasible with our computational resources.]
These basis set effects are fairly transferable to the other wave function methods that we have considered here.
This further motivates the ``pyramidal'' extrapolation scheme that we have employed to produce the TBE values (see Sec.~\ref{sec:TBE_compdet}).
Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the present context with much larger errors that the other schemes.
Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the present context with much larger errors than the other schemes.
Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns which makes SF-ADC(2.5) particularly accurate with a maximum error of \SI{0.029}{\eV} for the doubly-excited state {\twoAg}.
\alert{Due to the lack of dynamic correlation, CASSCF poorly describes the {\sBoneg} ionic state.
Although the autoisomerization barrier is not bad with the (4e,4o) active space, the {\sBoneg} is poorly describe mainly due to the poor CASSCF treatement that the second-order correction cannot fix.}
Although the automerization barrier is not bad with the (4e,4o) active space, the {\sBoneg} is poorly describe mainly due to the poor CASSCF treatement that the second-order correction cannot fix.}
Let us now move to discussion of the results of standard wave function methods that are reported in Table \ref{tab:D2h}.
Then we review the vertical energies obtained with multi-reference methods.
@ -883,7 +883,7 @@ The various TD-DFT functionals are not able to describe correctly the two single
\begin{squeezetable}
\begin{table*}
\caption{Energy differences between the various methods and the reference TBE values.
Note that AB stands for the autoisomerization barrier and is reported in \si{\kcalmol}.
Note that AB stands for the automerization 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 values between square brackets have been obtained by extrapolation via the procedure described in the corresponding footnote.}
\label{tab:TBE}
@ -945,7 +945,7 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
%================================================
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.
Table \ref{tab:TBE} shows the energy differences, for the automerization 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.
@ -1014,7 +1014,7 @@ The SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3)results are presented and are consiste
%%% TABLE I %%%
%\begin{squeezetable}
%\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 $\%T_1$ 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 automerization 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}
% \begin{ruledtabular}