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@ -247,13 +247,13 @@ A theoretical best estimate is defined for the autoisomerization barrier and for
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\label{sec:intro}
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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 been 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.
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Indeed, each computational model has its own theoretical and/or technical issues and the number of possible chemical scenario 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}
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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 been 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.
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Indeed, each computational model has its own theoretical and/or technical issues and the number of possible chemical scenario 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}
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Speaking of difficult task, 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.
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Speaking of difficult task, 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.
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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.
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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.
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This was confirmed by several experimental studies by Pettis and co-workers \cite{reeves_1969} and others. \cite{irngartinger_1983,ermer_1983,kreile_1986}
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This was confirmed by several experimental studies by Pettis and co-workers \cite{Reeves_1969} and others. \cite{Irngartinger_1983,Ermer_1983,Kreile_1986}
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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.
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However, at the {\Dfour} geometry, the {\sBoneg} ground state has two singly occupied frontier orbitals that are degenerate.
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@ -261,31 +261,31 @@ Therefore, one must take into account, at least, two electronic configurations t
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Of course, single-reference methods are naturally unable to describe such situations.
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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}).
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The autoisomerization barrier (AB) is defined as the difference between the square and rectangular ground-state energies.
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The energy of this barrier is estimated, experimentally, in the range of 1.6-10 \kcalmol \cite{whitman_1982} and previous multi-reference calculations yield an energy barrier in the range of 6-7 \kcalmol. \cite{eckert-maksic_2006}
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The energy of this barrier is estimated, experimentally, in the range of 1.6-10 \kcalmol \cite{Whitman_1982} and previous multi-reference calculations yield an energy barrier in the range of 6-7 \kcalmol. \cite{Eckert-Maksic_2006}
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%All these specificities of CBD make it a real playground for excited-states methods.
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The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}, where
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we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
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Due to the energy scale, the {\sBoneg}, {\twoAg} and {\Aoneg}, {\Btwog} states for the {\Dtwo} and {\Dfour} structures, respectively, are not shown.
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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 a challenge for adiabatic time-dependent density-functional theory (TD-DFT) \cite{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}
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Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown.
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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 a challenge 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}
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In order to tackle the problems of multi-configurational character and double excitations several ways are explored.
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The most evident route that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods.
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Among these methods, one can find complete active space self-consistent field (CASSCF) \cite{roos_1996}, the second perturbation-corrected variant (CASPT2) \cite{andersson_1990} and the second-order $n$-electron valence state perturbation theory (NEVPT2). \cite{angeli_2001,angeli_2001a,angeli_2002}
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The exponential scaling of these methods with the size of the active space is the limitation to the application of these ones to large molecules.
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In order to tackle the problem of multi-configurational character and double excitations, several ways can be explored.
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The most evident route is to rely on multiconfigurational methods.
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Among these methods, one can mention the complete active space self-consistent field (CASSCF) method, \cite{Roos_1996} its second perturbation-corrected variant (CASPT2) \cite{Andersson_1990,Andersson_1992,Roos_1995a} and the second-order $n$-electron valence state perturbation theory (NEVPT2) formalism. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
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The exponential scaling of these methods with the size of the active space is the principal limitation to their applicability to large molecules.
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Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism of coupled-cluster (CC) theory. \cite{hirata_2000,sundstrom_2014}
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However, to provide a correct description of doubly excited states one have to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{watson_2012}
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Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism \cite{Rowe_1968,Stanton_1993} of coupled-cluster (CC) theory. \cite{Kucharski_1991,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
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However, to provide a correct description of doubly-excited states, one have to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{Watson_2012,Loos_2018a,Loos_2020b}
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Again, due to the scaling of CC methods with the number of basis functions the applicability of these methods is limited to small molecules.
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An alternative to multiconfigurational and CC methods is the use of selected CI (SCI) methods for the computation of transition energies for singly and doubly excited states that are known to reach near full CI energies for small molecules.
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An alternative to multiconfigurational and CC methods is the use of selected CI (SCI) methods \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} for the computation of transition energies for singly and doubly excited states that are known to reach near full CI energies for small molecules. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021}
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These methods allow to avoid an exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space.
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Finally, to describe doubly excited states, one can think of spin-flip formalism established by Krylov in 2001.\cite{casanova_2020}
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Finally, to describe doubly excited states, one can think of spin-flip formalism established by Krylov in 2001.\cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020}
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To briefly introduce the spin-flip idea we can present it like: instead of taking the singlet ground state as reference, the reference configuration is taken as the lowest triplet state.
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So one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation.
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Obviously spin-flip methods have their own flaws, especially the spin contamination \cite{casanova_2020} (\ie, an artificial mixing of electronic states of different spin multiplicities) due to spin incompleteness of the spin-flip expansion and by spin contamination of the reference configuration.
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One can address part of this problem by expansion of the excitation order but with an increase of the computational cost or by complementing the spin-incomplete configuration set with the missing configurations.
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Obviously spin-flip methods have their own flaws, especially the spin contamination (\ie, an artificial mixing of electronic states of different spin multiplicities) due to spin incompleteness of the spin-flip expansion and by spin contamination of the reference configuration. \cite{Casanova_2020}
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One can address part of this problem by expansion of the excitation order but with an increase of the computational cost 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}
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In the present work we investigate {\oneAg}, {\tBoneg}, {\sBoneg}, {\twoAg} and {\sBoneg}, {\Atwog}, {\Aoneg},{\Btwog} excited states for the {\Dtwo} and {\Dfour} geometries, respectively.
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Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multiconfigurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods.
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@ -294,7 +294,8 @@ Section \ref{sec:res} is devoted to the discussion of our results, first we cons
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\begin{figure}
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\includegraphics[width=0.8\linewidth]{figure1.pdf}
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\caption{Pictorial representation of the ground and excited states of CBD and its properties under investigation.
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The singlet ground-state (S) and triplet (T) properties are represented in black and red, respectively.}
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The singlet ground-state (S) and triplet (T) properties are represented in black and red, respectively.
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The autoisomerization barrier (AB) is also represented.}
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\label{fig:CBD}
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\end{figure}
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@ -307,26 +308,26 @@ Section \ref{sec:res} is devoted to the discussion of our results, first we cons
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Selected configuration interaction calculations}
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\label{sec:SCI}
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States energies and excitations energies calculations in the SCI framework are performed using QUANTUM PACKAGE \cite{garniron_2019} where the CIPSI algorithm is implemented. The CIPSI algorithm allows to avoid the exponential increase of the CI expansion. To treat electronic states in the same way we use a state-averaged formalism meaning that the ground and excited states are represented with the same number and same set of determinants but using different CI coefficients. Then the SCI energy is the sum of two terms, the variational energy obtained by diagonalization of the CI matrix in the reference space and a second-order perturbative correction which estimates the contribution of the determinants not included in the CI space (estimate error in the truncation). It is possible to estimate the FCI limit for the total energies and compute the corresponding transition energies by extrapolating this second-order correction to zero. Extrapolation brings error and we can estimate this one by energy difference between excitation energies obtained with the largest SCI wave function and the FCI extrapolated value. These errors provide a rough idea of the quality of the FCI extrapolation and cannot be seen as true bar error, they are reported in the following tables.
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States energies and excitations energies calculations in the SCI framework are performed using QUANTUM PACKAGE \cite{Garniron_2019} where the CIPSI algorithm is implemented. The CIPSI algorithm allows to avoid the exponential increase of the CI expansion. To treat electronic states in the same way we use a state-averaged formalism meaning that the ground and excited states are represented with the same number and same set of determinants but using different CI coefficients. Then the SCI energy is the sum of two terms, the variational energy obtained by diagonalization of the CI matrix in the reference space and a second-order perturbative correction which estimates the contribution of the determinants not included in the CI space (estimate error in the truncation). It is possible to estimate the FCI limit for the total energies and compute the corresponding transition energies by extrapolating this second-order correction to zero. Extrapolation brings error and we can estimate this one by energy difference between excitation energies obtained with the largest SCI wave function and the FCI extrapolated value. These errors provide a rough idea of the quality of the FCI extrapolation and cannot be seen as true bar error, they are reported in the following tables.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Coupled-Cluster calculations}
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\label{sec:CC}
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Different flavours of coupled-cluster (CC) calculations are performed using different codes. Indeed, CC theory provides a hierarchy of methods that provide increasingly accurate energies via the increase of the maximum excitation degree of the cluster operator. Without any truncation of the cluster operator one has the full CC (FCC) that is equivalent to the full configuration interaction (FCI) giving the exact energy and wave function of the system for a fixed atomic basis set. However, due to the computational exponential scaling with system size we have to use truncated CC methods. The CC with singles and doubles (CCSD), CC with singles, doubles and triples (CCSDT) calculations are achieved with \textcolor{red}{CFOUR}. The calculations in the context of CC response theory or ``approximate'' series (CC3,CC4) are performed with \textcolor{red}{DALTON}.\cite{aidas_2014} The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \textcolor{red}{CFOUR} code. The CC2, \cite{christiansen_1995a,hattig_2000} CC3 \cite{christiansen_1995b,koch_1995} and CC4 \cite{kallay_2005,matthews_2020} methods can be seen as cheaper approximations of CCSD,\cite{purvis_1982} CCSDT \cite{noga_1987} and CCSDTQ \cite{kucharski_1991a} by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
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Different flavours of coupled-cluster (CC) calculations are performed using different codes. Indeed, CC theory provides a hierarchy of methods that provide increasingly accurate energies via the increase of the maximum excitation degree of the cluster operator. Without any truncation of the cluster operator one has the full CC (FCC) that is equivalent to the full configuration interaction (FCI) giving the exact energy and wave function of the system for a fixed atomic basis set. However, due to the computational exponential scaling with system size we have to use truncated CC methods. The CC with singles and doubles (CCSD), CC with singles, doubles and triples (CCSDT) calculations are achieved with \textcolor{red}{CFOUR}. The calculations in the context of CC response theory or ``approximate'' series (CC3,CC4) are performed with \textcolor{red}{DALTON}.\cite{Aidas_2014} The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \textcolor{red}{CFOUR} code. The CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020} methods can be seen as cheaper approximations of CCSD,\cite{Purvis_1982} CCSDT \cite{Noga_1987a} and CCSDTQ \cite{Kucharski_1991a} by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Multiconfigurational calculations}
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\label{sec:Multi}
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State-averaged complete-active-space self-consistent field (SA-CASSCF) calculations are performed with \textcolor{red}{MOLPRO}.\cite{werner_2012} On top of those, NEVPT2 calculations, both partially contracted (PC) and strongly contracted (SC) scheme are considered. The PC-NEVPT2 is theoretically more accurate to the strongly contracted version due to the larger number of perturbers and greater flexibility. CASPT2 is performed and extended multistate (XMS) CASPT2 for strong mixing between states with same spin and spatial symmetries is also performed.
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State-averaged complete-active-space self-consistent field (SA-CASSCF) calculations are performed with \textcolor{red}{MOLPRO}.\cite{Werner_2012} On top of those, NEVPT2 calculations, both partially contracted (PC) and strongly contracted (SC) scheme are considered. The PC-NEVPT2 is theoretically more accurate to the strongly contracted version due to the larger number of perturbers and greater flexibility. CASPT2 is performed and extended multistate (XMS) CASPT2 for strong mixing between states with same spin and spatial symmetries is also performed.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Spin-flip calculations}
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\label{sec:sf}
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In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore algebraic-diagrammatic construction \cite{schirmer_1982} (ADC) using standard ADC(2)-s \cite{trofimov_1997,dreuw_2015} and extended ADC(2)-x \cite{dreuw_2015} schemes as well as the ADC(3) \cite{dreuw_2015,trofimov_2002,harbach_2014} scheme. We also use spin-flip within the TD-DFT \cite{casida_1995} framework. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1. \cite{shao_2015} Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The B3LYP,\cite{becke_1988b,lee_1988a,becke_1993b} PBE0 \cite{adamo_1999a,ernzerhof_1999} and BH\&HLYP hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using range-separated hybrid (RSH) functionals as: CAM-B3LYP,\cite{yanai_2004a} LC-$\omega$PBE08 \cite{weintraub_2009a} and $\omega$B97X-V. \cite{mardirossian_2014}The main difference here between these RSH functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. To complete the use of spin-flip TD-DFT we also considered the hybrid meta-GGA functional M06-2X \cite{zhao_2008} and the RSH meta-GGA functional M11.\cite{peverati_2011} Note that all SF-TD-DFT calculations are done within the TDA approximation.
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In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore algebraic-diagrammatic construction \cite{Schirmer_1982} (ADC) using standard ADC(2)-s \cite{Trofimov_1997,Dreuw_2015} and extended ADC(2)-x \cite{Dreuw_2015} schemes as well as the ADC(3) \cite{Dreuw_2015,Trofimov_2002,Harbach_2014} scheme. We also use spin-flip within the TD-DFT \cite{Casida_1995} framework. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1. \cite{Shao_2015} Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The B3LYP,\cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using range-separated hybrid (RSH) functionals as: CAM-B3LYP,\cite{Yanai_2004a} LC-$\omega$PBE08 \cite{Weintraub_2009a} and $\omega$B97X-V. \cite{Mardirossian_2014}The main difference here between these RSH functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. To complete the use of spin-flip TD-DFT we also considered the hybrid meta-GGA functional M06-2X \cite{Zhao_2008} and the RSH meta-GGA functional M11.\cite{Peverati_2011} Note that all SF-TD-DFT calculations are done within the TDA approximation.
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%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -388,7 +389,7 @@ Two different sets of geometries obtained with different level of theory are con
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Angle between the \ce{C-H} bond and the \ce{C=C} bond.}
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\fnt[2]{From Ref.~\onlinecite{manohar_2008}.}
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\fnt[2]{From Ref.~\onlinecite{Manohar_2008}.}
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\end{table}
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\end{squeezetable}
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%%% %%% %%% %%%
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@ -816,7 +817,7 @@ For the {\sBoneg} state of the {\Dtwo} structure we see that all the xc-function
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Then, for the {\twoAg} state we obtain closer results of the SF-TD-DFT methods to the TBE than in the case of the {\sBoneg} state. Indeed, we have an energy difference of about \SIrange{0.01}{0.34}{\eV} for the {\twoAg} state whereas we have \SIrange{0.35}{0.93}{\eV} for the {\sBoneg} state. The ADC schemes give the same error to the TBE value than for the other singlet state with \SI{0.02}{\eV} for the ADC(2) scheme and \SI{0.07}{\eV} for the ADC(3) one. The ADC(2)-x scheme provides a larger error with \SI{0.45}{\eV} of energy difference. Here, the CC methods manifest more variations with \SI{0.63}{\eV} for the CC3 value and \SI{0.28}{\eV} for the CCSDT compared to the TBE values. The CC4 method provides a small error with less than 0.01 eV of energy difference. The multiconfigurational methods globally give smaller error than for the other singlet state with, for the two active spaces, \SIrange{0.03}{0.12}{\eV} compared to the TBE value. We can notice that CC3 and CCSDT provide larger energy errors for the {\twoAg} state than for the {\sBoneg} state, this is due to the strong multiconfigurational character of the {\twoAg} state whereas the {\sBoneg} state has a very weak multiconfigurational character. It is interesting to see that SF-TD-DFT and SF-ADC methods give better results compared to the TBE than CC3 and even CCSDT meaning that spin-flip methods are able to describe double excited states. Note that multireference methods obviously give better results too for the {\twoAg} state.
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Finally we look at the vertical energy errors for the \Dfour structure. First, we consider the {\Atwog} state, the SF-TD-DFT methods give errors of about \SIrange{0.07}{1.6}{\eV} where the largest energy differences are provided by the hybrid functionals. The ADC schemes give similar error with around \SIrange{0.06}{1.1}{\eV} of energy difference. For the CC methods we have an energy error of \SI{0.06}{\eV} for CCSD and less than \SI{0.01}{\eV} for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) \SI{0.29}{\eV} of error and \SI{0.02}{\eV} for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about \SIrange{0.12}{0.13}{\eV}. A larger active space shows again an improvement with \SI{0.23}{\eV} of error for CASSCF(12,12) and around \SIrange{0.01}{0.04}{\eV} for the other multireference methods. CIPSI provides similar error with \SI{0.02}{\eV}. Then, we look at the {\Aoneg} state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about \SIrange{0.10}{1.03}{\eV}. The ADC schemes give better errors with around \SIrange{0.07}{0.41}{\eV} and where again the ADC(2)-x does not improve the ADC(2)-s energy but also gives worse results. For the CC methods we have energy errors of about \SIrange{0.10}{0.16}{\eV} and CC4 provides really close energy to the TBE one with \SI{0.01}{\eV} of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of \SIrange{0.01}{0.73}{\eV} and \SIrange{0.02}{0.44}{\eV} respectively with the largest errors coming from the CASSCF method. Lastly, we look at the {\Btwog} state where we have globally larger errors. The SF-TD-DFT exhibits errors of \SIrange{0.43}{1.50}{\eV} whereas ADC schemes give errors of \SIrange{0.18}{0.30}{\eV}. CC3 and CCSDT provide energy differences of \SIrange{0.50}{0.69}{\eV} and the CC4 shows again close energy to the CCSDTQ TBE energy with \SI{0.01}{\eV} of error. The multireference methods give energy differences of \SIrange{0.38}{1.39}{\eV} and \SIrange{0.11}{0.60}{\eV} for the four by four and twelve by twelve active spaces respectively. We can notice that we have larger variations for the vertical energies of the square CBD than for the rectangular one. This can be explained by the fact that because of the degeneracy in the {\Dfour} structure it leads to strong multiconfigurational character states where single reference methods are unreliable. We can also see that for the CC methods we have a better description of the {\Aoneg} state than the {\Btwog} state, this can be related, as previously said in Subsubsection \ref{sec:D4h}, to the fact that {\Btwog} corresponds to a double excitation from the reference state. To obtain an improved description of the {\Btwog} state we have to include quadruples. At the end of Table \ref{tab:TBE} we show some literature results obtain from Ref.~\onlinecite{lefrancois_2015,manohar_2008} where the cc-pVTZ basis is used. The SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3)results are presented and are consistent with our results with the exact same schemes but with the aug-cc-pVTZ basis.
|
||||
Finally we look at the vertical energy errors for the \Dfour structure. First, we consider the {\Atwog} state, the SF-TD-DFT methods give errors of about \SIrange{0.07}{1.6}{\eV} where the largest energy differences are provided by the hybrid functionals. The ADC schemes give similar error with around \SIrange{0.06}{1.1}{\eV} of energy difference. For the CC methods we have an energy error of \SI{0.06}{\eV} for CCSD and less than \SI{0.01}{\eV} for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) \SI{0.29}{\eV} of error and \SI{0.02}{\eV} for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about \SIrange{0.12}{0.13}{\eV}. A larger active space shows again an improvement with \SI{0.23}{\eV} of error for CASSCF(12,12) and around \SIrange{0.01}{0.04}{\eV} for the other multireference methods. CIPSI provides similar error with \SI{0.02}{\eV}. Then, we look at the {\Aoneg} state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about \SIrange{0.10}{1.03}{\eV}. The ADC schemes give better errors with around \SIrange{0.07}{0.41}{\eV} and where again the ADC(2)-x does not improve the ADC(2)-s energy but also gives worse results. For the CC methods we have energy errors of about \SIrange{0.10}{0.16}{\eV} and CC4 provides really close energy to the TBE one with \SI{0.01}{\eV} of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of \SIrange{0.01}{0.73}{\eV} and \SIrange{0.02}{0.44}{\eV} respectively with the largest errors coming from the CASSCF method. Lastly, we look at the {\Btwog} state where we have globally larger errors. The SF-TD-DFT exhibits errors of \SIrange{0.43}{1.50}{\eV} whereas ADC schemes give errors of \SIrange{0.18}{0.30}{\eV}. CC3 and CCSDT provide energy differences of \SIrange{0.50}{0.69}{\eV} and the CC4 shows again close energy to the CCSDTQ TBE energy with \SI{0.01}{\eV} of error. The multireference methods give energy differences of \SIrange{0.38}{1.39}{\eV} and \SIrange{0.11}{0.60}{\eV} for the four by four and twelve by twelve active spaces respectively. We can notice that we have larger variations for the vertical energies of the square CBD than for the rectangular one. This can be explained by the fact that because of the degeneracy in the {\Dfour} structure it leads to strong multiconfigurational character states where single reference methods are unreliable. We can also see that for the CC methods we have a better description of the {\Aoneg} state than the {\Btwog} state, this can be related, as previously said in Subsubsection \ref{sec:D4h}, to the fact that {\Btwog} corresponds to a double excitation from the reference state. To obtain an improved description of the {\Btwog} state we have to include quadruples. At the end of Table \ref{tab:TBE} we show some literature results obtain from Ref.~\onlinecite{Lefrancois_2015,Manohar_2008} where the cc-pVTZ basis is used. The SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3)results are presented and are consistent with our results with the exact same schemes but with the aug-cc-pVTZ basis.
|
||||
|
||||
|
||||
%Here again we can make the same comment for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states of the square CBD than the $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states of the rectangular CBD. The first state ($2\,{}^1A_{1g}$) has a strong multiconfigurational character
|
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@ -931,10 +932,10 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
|
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\fnt[2]{Value obtained using the NEVPT2(12,12) one.}
|
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\fnt[3]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.}
|
||||
\fnt[4]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ.}
|
||||
\fnt[5]{Value obtained from Ref.~\onlinecite{lefrancois_2015} at the SF-ADC(2)-s/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[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[5]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-s/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[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{squeezetable}
|
||||
|
||||
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Reference in New Issue
Block a user