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6
Manuscript/CBD-SI.tex
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@ -190,7 +190,7 @@ CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
\bf{TBE} & $[\bf{8.93}]$\fnm[1] & $[\bf{1.462}]$\fnm[2] & $[\bf{3.125}]$\fnm[3] & $[\bf{4.149}]$\fnm[3] & $[\bf{0.144}]$\fnm[4] & $[\bf{1.500}]$\fnm[3] & $[\bf{2.034}]$\fnm[3] \\[0.1cm] \bf{TBE} & $[\bf{8.93}]$\fnm[1] & $[\bf{1.462}]$\fnm[2] & $[\bf{3.125}]$\fnm[3] & $[\bf{4.149}]$\fnm[3] & $[\bf{0.144}]$\fnm[4] & $[\bf{1.500}]$\fnm[3] & $[\bf{2.034}]$\fnm[3] \\[0.1cm]
Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] & $0.266$\fnm[5] & $1.664$\fnm[5] & $1.910$\fnm[5] \\ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] & $0.266$\fnm[5] & $1.664$\fnm[5] & $1.910$\fnm[5] \\
& $10.35$\fnm[6] & $1.576$\fnm[6] & $3.141$\fnm[6] & $3.796$\fnm[6] & $0.217$\fnm[6] & $1.123$\fnm[6] & $1.799$\fnm[6]\\ & $10.35$\fnm[6] & $1.576$\fnm[6] & $3.141$\fnm[6] & $3.796$\fnm[6] & $0.217$\fnm[6] & $1.123$\fnm[6] & $1.799$\fnm[6]\\
& $9.58$ \fnm[7]& $1.456$\fnm[7] & $3.285$\fnm[7] & $4.334$\fnm[7] & $0.083$\fnm[7] & $1.621$\fnm[7] & $1.930$\fnm[7] \\ & $9.58$\fnm[7]& $1.456$\fnm[7] & $3.285$\fnm[7] & $4.334$\fnm[7] & $0.083$\fnm[7] & $1.621$\fnm[7] & $1.930$\fnm[7] \\
& $7.48$\fnm[8]& $1.654$\fnm[8] & $3.416$\fnm[8] & $4.360$\fnm[8] & $0.369$\fnm[8] & $1.824$\fnm[8] & $2.143$\fnm[8] \\ & $7.48$\fnm[8]& $1.654$\fnm[8] & $3.416$\fnm[8] & $4.360$\fnm[8] & $0.369$\fnm[8] & $1.824$\fnm[8] & $2.143$\fnm[8] \\
\end{tabular} \end{tabular}
@ -209,7 +209,7 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
%%% %%% %%% %%% %%% %%% %%% %%%
\begin{table} \begin{table}
\caption{} \caption{Automerization energy (in \si{\kcalmol}) of CBD computed at various levels of theory.}
% \label{} % \label{}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcr} \begin{tabular}{lcr}
@ -247,7 +247,7 @@ Literature & $8.53$\fnm[5] & $1.573$\fnm[5] & $3.208$\fnm[5] & $4.247$\fnm[5] &
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrrrrrr} \begin{tabular}{lrrrrrr}
&\mc{2}{r}{$\expval*{S^2}$ ({\Dtwo})} & \mc{3}{r}{{$\expval*{S^2}$ (\Dfour})} \\ &\mc{3}{c}{$\expval*{S^2}$ for {\Dtwo} geometry} & \mc{3}{c}{{$\expval*{S^2}$ for {\Dfour} geometry}} \\
\cline{2-4} \cline{5-7} \cline{2-4} \cline{5-7}
Method & {\tBoneg} & {\sBoneg} & {\twoAg} & {\Atwog} & {\Aoneg} & {\Btwog} \\ Method & {\tBoneg} & {\sBoneg} & {\twoAg} & {\Atwog} & {\Aoneg} & {\Btwog} \\
\hline \hline

32
Manuscript/CBD.tex
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@ -132,7 +132,7 @@ In this context, an interesting alternative to multi-configurational and CC meth
For example, the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method limits the 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. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2017,Garniron_2018,Garniron_2019} For example, the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method limits the 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. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2017,Garniron_2018,Garniron_2019}
Nonetheless, SCI methods remain very expensive and can be applied to a limited number of situations. Nonetheless, SCI methods remain very expensive and can be applied to a limited number of situations.
Finally, another option to deal with these chemical scenarios is to rely on the cheaper spin-flip formalism, established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020} where one accesses the ground and doubly-excited states via a single (spin-flip) excitation and deexcitation from the lowest triplet state, respectively. Finally, another option to deal with these chemical scenarios is to rely on the cheaper spin-flip formalism, established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020} where one accesses the ground and doubly-excited states via a single (spin-flip) de-excitation and excitation from the lowest triplet state, respectively.
Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, the artificial mixing of electronic states with different spin multiplicities) due principally to spin incompleteness in the spin-flip expansion but also to the spin contamination of the reference configuration. \cite{Casanova_2020} Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, the artificial mixing of electronic states with different spin multiplicities) due principally to spin incompleteness in the spin-flip expansion but also to the spin contamination of the reference configuration. \cite{Casanova_2020}
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} 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. both solutions being associated with an additional computational cost.
@ -178,14 +178,14 @@ This type of extrapolation procedures is now routine in SCI methods as well as o
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:\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. 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 high-order amplitudes.
Here, we have performed CC calculations using various codes. Here, we have performed CC calculations using various codes.
Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achieved with CFOUR, \cite{Matthews_2020} with which only singlet excited states can be computed. Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achieved with CFOUR, \cite{Matthews_2020} with which only singlet excited states can be computed.
In some cases, we have also computed (singlet and triplet) excitation energies and properties (such as the percentage of single excitations involved in a given transition, namely $\%T_1$) at the CC3 level with DALTON \cite{Aidas_2014} and CCSDT level with MRCC. \cite{mrcc} In some cases, we have also computed (singlet and triplet) excitation energies and properties (such as the percentage of single excitations involved in a given transition, namely $\%T_1$) at the CC3 level with DALTON \cite{Aidas_2014} and at the CCSDT level with MRCC. \cite{mrcc}
To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell or unrestricted formalisms, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state {\Aoneg} as reference. To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell or unrestricted formalisms, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state {\Aoneg} as reference.
Hence, the open-shell ground state {\sBoneg} and the {\Btwog} state appear as a deexcitation and an excitation, respectively. Hence, the open-shell ground state {\sBoneg} and the {\Btwog} state appear as a de-excitation and an excitation, respectively.
With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below). With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -194,7 +194,7 @@ With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character,
\label{sec:Multi} \label{sec:Multi}
State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020} State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020}
For each excited state, a set of state-averaged orbitals is computed by taking into account the excited state of interest as well as the ground state (even if it has a different symmetry). For each excited state, a set of state-averaged orbitals is computed by taking into account the excited state of interest as well as the ground state (even if it has a different symmetry).
Two active spaces have been considered: (i) a minimal (4e,4o) active space including valence $\pi$ orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the $\sigma_\text{CC}$ and $\sigma_\text{CC}^*$ orbitals. Two active spaces have been considered: (i) a minimal (4e,4o) active space including the valence $\pi$ orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the $\sigma_\text{CC}$ and $\sigma_\text{CC}^*$ orbitals.
For ionic states, like the {\sBoneg} state of CBD, it is particularly important to take into account the $\sigma$-$\pi$ coupling. \cite{Davidson_1996,Angeli_2009,BenAmor_2020} For ionic states, like the {\sBoneg} state of CBD, it is particularly important to take into account the $\sigma$-$\pi$ coupling. \cite{Davidson_1996,Angeli_2009,BenAmor_2020}
On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002} On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
@ -209,7 +209,7 @@ All these calculations are also carried out with MOLPRO. \cite{Werner_2020}
\subsection{Spin-flip calculations} \subsection{Spin-flip calculations}
\label{sec:sf} \label{sec:sf}
Within the spin-flip formalism, one considers the lowest triplet state as reference instead of the singlet ground state. Within the spin-flip formalism, one considers the lowest triplet state as reference instead of the singlet ground state.
Ground-state energies are then computed as sums of the triplet reference state energy and the corresponding deexcitation energy. Ground-state energies are then computed as sums of the triplet reference state energy and the corresponding de-excitation energy.
Likewise, excitation energies with respect to the singlet ground state are computed as differences of excitation energies with respect to the reference triplet state. Likewise, excitation energies with respect to the singlet ground state are computed as differences of excitation energies with respect to the reference triplet state.
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).
@ -228,7 +228,7 @@ Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximat
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theoretical best estimates} \subsection{Theoretical best estimates}
\label{sec:TBE_compdet} \label{sec:TBE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
When technically possible, each level of theory is tested with four Gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989} When technically possible, each level of theory is tested with four Gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989}
This helps us to assess the convergence of each property with respect to the size of the basis set. This helps us to assess the convergence of each property with respect to the size of the basis set.
@ -299,15 +299,15 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained at th
State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & $\angle\,\ce{H-C=C}$ \\ 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[1] & 1.354 & 1.566 & 1.077 & 134.99 \\ CASPT2(12,12)/aug-cc-pVTZ\fnm[1] & 1.354 & 1.566 & 1.077 & 134.99 \\
&CC3/aug-cc-pVTZ \fnm[1] & 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[2] & 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[1] & 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[1] & 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[1] & 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[2] & 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]{This work.} \fnt[1]{This work.}
@ -619,7 +619,7 @@ Second, we discuss the various SF-ADC schemes (Table \ref{tab:sf_D2h}), \ie, SF-
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}. 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. 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}). This further motivates the ``pyramidal'' extrapolation scheme that we have employed to produce the TBE values (see Sec.~\ref{sec:TBE}).
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. 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}. 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}.
@ -831,7 +831,7 @@ The (12e,12o) active space significantly alleviate these effects, and, as usuall
Finally, let us consider the excitation energies computed with various CC models and gathered in Table \ref{tab:D4h}. Finally, let us consider the excitation energies computed with various CC models and gathered in Table \ref{tab:D4h}.
As mentioned in Sec.~\ref{sec:CC}, we remind the reader that these calculations are performed by considering the {\Aoneg} state as reference, and that, therefore, As mentioned in Sec.~\ref{sec:CC}, we remind the reader that these calculations are performed by considering the {\Aoneg} state as reference, and that, therefore,
{\sBoneg} and {\Btwog} are obtained as a deexcitation and an excitation, respectively. {\sBoneg} and {\Btwog} are obtained as a de-excitation and an excitation, respectively.
Consequently, with respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character. Consequently, with respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character.
This explains why one observes a slower convergence of the transition energies in the case of {\sBoneg} as shown in Fig.~\ref{fig:D4h}. This explains why one observes a slower convergence of the transition energies in the case of {\sBoneg} as shown in Fig.~\ref{fig:D4h}.
It is clear from the results of Table \ref{tab:D4h} that, if one wants to reach high accuracy, it is mandatory to include quadruple excitations. It is clear from the results of Table \ref{tab:D4h} that, if one wants to reach high accuracy, it is mandatory to include quadruple excitations.