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@ -85,8 +85,8 @@
\begin{abstract} \begin{abstract}
The cyclobutadiene molecule is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods. 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. 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 $D_{4h}$ arrangements. 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 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. 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)]. 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 autoisomerization barrier and for each vertical transition energy.
\end{abstract} \end{abstract}
@ -102,20 +102,19 @@ Despite the fact that excited states are involved in ubiquitous processes such a
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_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}
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. 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.
The simple H\"uckel molecular orbital theory (wrongly) predicts a triplet ground state at the {\Dfour} square geometry, with two singly-occupied frontier orbitals that are degenerate by symmetry (Hund's rule), while state-of-the-art \textit{ab initio} methods (correctly) predict an open-shell singlet ground state. 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.
This degeneracy is lifted by the so-called Jahn-Teller effect, \ie, by a descent in symmetry (from {\Dfour} to {\Dtwo} point group) via a geometrical distortion of the molecule, leading to a closed-shell singlet ground state (see below). This degeneracy is lifted by the so-called Jahn-Teller effect, \ie, by a descent in symmetry (from {\Dfour} to {\Dtwo} point group) via a geometrical distortion of the molecule, leading to a closed-shell singlet ground state (see below).
This was confirmed by several experimental studies by Pettis and co-workers \cite{Reeves_1969} and others. \cite{Irngartinger_1983,Ermer_1983,Kreile_1986} This was confirmed by several experimental studies by Pettis and co-workers \cite{Reeves_1969} and others. \cite{Irngartinger_1983,Ermer_1983,Kreile_1986}
At the {\Dtwo} geometry, the {\oneAg} ground state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods. In the {\Dtwo} symmetry, the {\oneAg} ground state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods.
However, at the {\Dfour} geometry, the {\sBoneg} ground state has two singly occupied frontier orbitals that are degenerate. However, in the {\Dfour} symmetry, the {\sBoneg} ground state has two singly occupied frontier orbitals that are degenerate.
Therefore, one must take into account, at least, two electronic configurations to properly model this multi-configurational scenario. 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. Of course, single-reference methods are naturally unable to describe such situations.
The singlet ground state 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 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}
%All these specificities of CBD make it a real playground for excited-states methods.
The lowest-energy excited states of CBD in both symetries 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.
Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be an absolute nightmare for adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{Christiansen_1995,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles, and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004} Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be an absolute nightmare for adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{Christiansen_1995,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles, and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
@ -180,7 +179,7 @@ This type of extrapolation procedures is now routine in SCI methods as well as o
\subsection{Coupled-cluster calculations} \subsection{Coupled-cluster calculations}
\label{sec:CC} \label{sec:CC}
Coupled-cluster theory provides a hierarchy of methods that yields increasingly accurate ground state energies by ramping up the maximum excitation degree of the cluster operator: \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002b,Bartlett_2007,Shavitt_2009} CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1991a,Kucharski_1992} etc. Coupled-cluster theory provides a hierarchy of methods that yields increasingly accurate ground state energies by ramping up the maximum excitation degree of the cluster operator: \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002b,Bartlett_2007,Shavitt_2009} CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1991a,Kucharski_1992} etc.
As mentioned above, CC theory can be extended to excited states via the EOM formalism, \cite{Rowe_1968,Stanton_1993} where one diagonalizes the similarity-transformed Hamiltonian in a CI basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states: EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc.\cite{Noga_1987a,Koch_1990b,Kucharski_1991,Stanton_1993,Christiansen_1998,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004} As mentioned above, CC theory can be extended to excited states via the EOM formalism, \cite{Rowe_1968,Stanton_1993} where one diagonalizes the similarity-transformed Hamiltonian in a CI basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states:\cite{Noga_1987a,Koch_1990b,Kucharski_1991,Stanton_1993,Christiansen_1998,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004} EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc.
In the following, we will omit the prefix EOM for the sake of conciseness. In the following, we will omit the prefix EOM for the sake of conciseness.
Alternatively to the ``complete'' CC models, one can also employed the CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020,Loos_2021} methods which can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes. Alternatively to the ``complete'' CC models, one can also employed the CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020,Loos_2021} methods which can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
@ -289,27 +288,26 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained at th
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{Optimized geometries associated with several states of CBD computed with various levels of theory. \caption{Optimized geometries associated with several states of CBD computed with various levels of theory.
Bond lengths are in \si{\angstrom} and angles are in degree.} Bond lengths are in \si{\angstrom} and angles ($\angle$) are in degree.}
\label{tab:geometries} \label{tab:geometries}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lllrrr} \begin{tabular}{lllrrr}
State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & \ce{H-C=C}\fnm[1] \\ State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & $\angle\,\ce{H-C=C}$ \\
\hline \hline
{\Dtwo} ({\oneAg}) & {\Dtwo} ({\oneAg}) &
CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.355 & 1.566 & 1.077 & 134.99 \\ CASPT2(12,12)/aug-cc-pVTZ \fnm[1] & 1.355 & 1.566 & 1.077 & 134.99 \\
&CC3/aug-cc-pVTZ \fnm[2] & 1.344 & 1.565 & 1.076 & 135.08 \\ &CC3/aug-cc-pVTZ \fnm[1] & 1.344 & 1.565 & 1.076 & 135.08 \\
&CCSD(T)/cc-pVTZ \fnm[3] & 1.343 & 1.566 & 1.074 & 135.09\\ &CCSD(T)/cc-pVTZ \fnm[2] & 1.343 & 1.566 & 1.074 & 135.09\\
{\Dfour} ({\sBoneg}) & {\Dfour} ({\sBoneg}) &
CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.449 & 1.449 & 1.076 & 135.00 \\ CASPT2(12,12)/aug-cc-pVTZ \fnm[1] & 1.449 & 1.449 & 1.076 & 135.00 \\
{\Dfour} ({\Atwog}) & {\Dfour} ({\Atwog}) &
CASPT2(12,12)/aug-cc-pVTZ \fnm[2] & 1.445 & 1.445 & 1.076 & 135.00 \\ CASPT2(12,12)/aug-cc-pVTZ \fnm[1] & 1.445 & 1.445 & 1.076 & 135.00 \\
&RO-CCSD(T)/aug-cc-pVTZ \fnm[2] & 1.439 & 1.439 & 1.075 & 135.00\\ &RO-CCSD(T)/aug-cc-pVTZ \fnm[1] & 1.439 & 1.439 & 1.075 & 135.00\\
&RO-CCSD(T)/cc-pVTZ \fnm[3] & 1.439 & 1.439 & 1.073 & 135.00\\ &RO-CCSD(T)/cc-pVTZ \fnm[2] & 1.439 & 1.439 & 1.073 & 135.00\\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Angle between the \ce{C-H} bond and the \ce{C=C} bond.} \fnt[1]{This work.}
\fnt[2]{This work.} \fnt[2]{From Ref.~\onlinecite{Manohar_2008}.}
\fnt[3]{From Ref.~\onlinecite{Manohar_2008}.}
\end{table} \end{table}
\end{squeezetable} \end{squeezetable}
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