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donw for titou for now

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Pierre-Francois Loos 2022-04-06 18:23:13 +02:00
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Manuscript/CBD.tex
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@ -375,7 +375,7 @@ CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[\bf 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\
%%% FIGURE II %%% %%% FIGURE II %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{AB_AVTZ} \includegraphics[width=0.8\linewidth]{AB_AVTZ}
\caption{Automerization 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.
See {\SupInf} for the raw data.} See {\SupInf} for the raw data.}
\label{fig:AB} \label{fig:AB}
@ -390,13 +390,13 @@ Indeed, hybrid functionals with a large fraction of short-range exact exchange (
However, they are still off by \SIrange{1}{4}{\kcalmol} from the TBE reference value. However, they are still off by \SIrange{1}{4}{\kcalmol} from the TBE reference value.
For the RSH functionals, the automerization 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} 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. 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. 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 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. 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). 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). 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. Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models.
Concerning the multi-reference approaches with the minimal (4e,4o) active space, the TBEs are bracketed by the CASPT2 and NEVPT2 values that differ by approximately \SI{1.5}{\kcalmol} for all bases. Concerning the multi-reference approaches with the minimal (4e,4o) active space, the TBEs are bracketed by the CASPT2 and NEVPT2 values that differ by approximately \SI{1.5}{\kcalmol} for all bases.
In this case, the NEVPT2 values are fairly accurate with differences below half a \si{\kcalmol} compared to the TBEs. In this case, the NEVPT2 values are fairly accurate with differences below half a \si{\kcalmol} compared to the TBEs.
@ -407,7 +407,7 @@ For each basis set, both CASPT2(12,12) and NEVPT2(12,12) are less than a \si{\kc
For the two active spaces that we have considered here, the PC- and SC-NEVPT2 schemes provide nearly identical barriers independently of the size of the one-electron basis. For the two active spaces that we have considered here, the PC- and SC-NEVPT2 schemes provide nearly identical barriers independently of the size of the one-electron basis.
Finally, for the CC family of methods, we observe the usual systematic improvement following the series CCSD $<$ CC3 $<$ CCSDT $<$ CC4 $<$ CCSDTQ, which is also linked to their increase in computational cost: $\order*{N^6}$, $\order*{N^7}$, $\order*{N^8}$, $\order*{N^9}$, and $\order*{N^{10}}$, respectively. Finally, for the CC family of methods, we observe the usual systematic improvement following the series CCSD $<$ CC3 $<$ CCSDT $<$ CC4 $<$ CCSDTQ, which is also linked to their increase in computational cost: $\order*{N^6}$, $\order*{N^7}$, $\order*{N^8}$, $\order*{N^9}$, and $\order*{N^{10}}$, respectively.
Note that the introduction of the triple excitations is clearly mandatory to have an accuracy beyond SF-TD-DFT, while it is also clear that the iterative triples and quadruples can be included approximately via the CC3 and CC4 methods. Note that the introduction of the triple excitations is clearly mandatory to have an accuracy beyond SF-TD-DFT, while it is also clear that the iterative triples and quadruples can be included approximately via the CC3 and CC4 methods, respectively.
%================================================ %================================================
%================================================ %================================================
@ -581,7 +581,7 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
%%% FIGURE III %%% %%% FIGURE III %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{D2h} \includegraphics[width=0.8\linewidth]{D2h}
\caption{Vertical excitation energies of the {\tBoneg}, {\sBoneg}, and {\twoAg} states at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state using the aug-cc-pVTZ basis. \caption{Vertical excitation energies of the {\tBoneg}, {\sBoneg}, and {\twoAg} states at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state using the aug-cc-pVTZ basis.
See {\SupInf} for the raw data.} See {\SupInf} for the raw data.}
%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.} %Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
@ -596,7 +596,7 @@ At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in th
Therefore, the two formers are dominated by single excitations, while the latter state is a genuine double excitation. Therefore, the two formers are dominated by single excitations, while the latter state is a genuine double excitation.
First, let us discuss basis set effects at the SF-TD-DFT level (Table \ref{tab:sf_D2h}). First, let us discuss basis set effects at the SF-TD-DFT level (Table \ref{tab:sf_D2h}).
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} As expected, these are found to be small and the results are basically converged to the complete 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. 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 fraction 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} Functionals with large fraction 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. 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.
@ -627,13 +627,13 @@ Let us now move to the discussion of the results concerning standard wave functi
Regarding the multi-configurational calculations, the most striking result is the poor description of the {\sBoneg} ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state. Regarding the multi-configurational calculations, the most striking result is the poor description of the {\sBoneg} ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state.
Of course, the PT2 correction is able to correct the root-flipping problem but cannot provide quantitative excitation energies due to the poor zeroth-order treatment. Of course, the PT2 correction is able to correct the root-flipping problem but cannot provide quantitative excitation energies due to the poor zeroth-order treatment.
Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}. Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}.
This feature is characteristic of the inadequacy of the active space. This feature is characteristic of the inadequacy of the active space to model such a state.
For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller and typically below \SI{0.1}{\eV}. For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller and typically below \SI{0.1}{\eV}.
Using a larger active resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet state) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis. Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet state) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4. Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4.
For the doubly-excited state, {\twoAg}, the convergence of the CC expansion is much slower but it is worth pointing out that the inclusion of approximate quadruples via CC4 is particularly effective in the present case. For the doubly-excited state, {\twoAg}, the convergence of the CC expansion is much slower but it is worth pointing out that the inclusion of approximate quadruples via CC4 is particularly effective in the present case.
The CCSDTQ excitation energies (which are used to define the TBEs) are systematically within the error bar of the CIPSI calculations, which confirms the outstanding performance of CC methods including quadruple excitations in the context of excited states. The CCSDTQ excitation energies (which are used to define the TBEs) are systematically within the error bar of the CIPSI calculations, which confirms the outstanding performance of CC methods that include quadruple excitations in the context of excited states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{{\Dfour} square-planar geometry} \subsubsection{{\Dfour} square-planar geometry}
@ -791,7 +791,7 @@ CIPSI & 6-31+G(d) & $0.201\pm 0.003$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
%%% FIGURE IV %%% %%% FIGURE IV %%%
\begin{figure*} \begin{figure*}
\includegraphics[width=\linewidth]{D4h} \includegraphics[width=0.8\linewidth]{D4h}
\caption{Vertical excitation energies (in \si{\eV}) of the {\Atwog}, {\Aoneg}, and {\Btwog} states at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state using the aug-cc-pVTZ basis. \caption{Vertical excitation energies (in \si{\eV}) of the {\Atwog}, {\Aoneg}, and {\Btwog} states at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state using the aug-cc-pVTZ basis.
See {\SupInf} for the raw data.} See {\SupInf} for the raw data.}
%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.} %Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
@ -807,10 +807,10 @@ However, it is clear from the inspection of the wave function that, with respect
As for the previous geometry we start by discussing the SF-TD-DFT results (Table \ref{tab:sf_D4h}), and in particular the singlet-triplet gap, \ie, the energy difference between {\sBoneg} and {\Atwog}. As for the previous geometry we start by discussing the SF-TD-DFT results (Table \ref{tab:sf_D4h}), and in particular the singlet-triplet gap, \ie, the energy difference between {\sBoneg} and {\Atwog}.
For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the TBE value is \SI{0.144}{\eV}) but it is worth mentioning that B3LYP and PBE0 predict a negative singlet-triplet gap (hence a triplet ground state). For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the TBE value is \SI{0.144}{\eV}) but it is worth mentioning that B3LYP and PBE0 predict a negative singlet-triplet gap (hence a triplet ground state).
Increasing the fraction of exact exchange in hybrids or relying on RSHs (even with a small amount of short-range exact exchange) allows to recover a positive gap and a singlet ground state. Increasing the fraction of exact exchange in hybrids or relying on RSHs (even with a small amount of short-range exact exchange) allows to recover a positive gap and a singlet ground state.
At the SF-TD-DFT level, the energy gap between the two singlet excited states, {\Aoneg} and {\Btwog}, is particularly small and grows moderately with the amount of short-range exact exchange. At the SF-TD-DFT level, the energy gap between the two singlet excited states, {\Aoneg} and {\Btwog}, is particularly small and grows moderately with the amount of exact exchange at short range.
The influence of the exact exchange on the singlet energies is quite significant with an energy difference of the order of \SI{1}{\eV} between the functional with the smallest amount of exact exchange (B3LYP) and the functional with the largest amount (M06-2X). The influence of the exact exchange on the singlet energies is quite significant with an energy difference of the order of \SI{1}{\eV} between the functional with the smallest amount of exact exchange (B3LYP) and the functional with the largest amount (M06-2X).
As for the excitation energies computed at the {\Dtwo} ground-state equilibrium structure and the automerization barrier, functionals with a large fraction of short-range exact exchange deliver much more accurate results. As for the excitation energies computed at the {\Dtwo} ground-state equilibrium structure and the automerization barrier, functionals with a large fraction of short-range exact exchange deliver much more accurate results.
Yet, the transition energy to {\Btwog} is off by more than half an \si{\eV} compared to the TBE, while the doubly-excited state is much closer to the reference value (errors of \SI{-0.251}{}, \SI{-0.097}{}, and \SI{-0.312}{\eV} for BH\&HLYP, M06-2X, and M11, respectively) Yet, the transition energy to {\Btwog} is off by more than half an \si{\eV} compared to the TBE, while the doubly-excited state is much closer to the reference value (errors of \SI{-0.251}{}, \SI{-0.097}{}, and \SI{-0.312}{\eV} for BH\&HLYP, M06-2X, and M11, respectively).
Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level. Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level.
Note again that the $\expval*{S^2}$ values reported in {\SupInf} show that there is no significant spin contamination in these excited states. Note again that the $\expval*{S^2}$ values reported in {\SupInf} show that there is no significant spin contamination in these excited states.
@ -844,13 +844,13 @@ As a final comment, we can note that the agreement between our CCSDTQ-based TBEs
\label{sec:conclusion} \label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the present study, we have benchmarked a larger number of computational methods on the automerization barrier and the vertical energies of the cyclobutadiene (CBD) molecule in its square ({\Dfour}) and rectangular ({\Dtwo}) arrangements, for which we have defined theoretical best estimates (TBEs) based on extrapolated CCSDTQ/aug-cc-pVTZ data. In the present study, we have benchmarked a larger number of computational methods on the automerization barrier and the vertical excitation energies of the cyclobutadiene (CBD) molecule in its square ({\Dfour}) and rectangular ({\Dtwo}) arrangements, for which we have defined theoretical best estimates (TBEs) based on extrapolated CCSDTQ/aug-cc-pVTZ data.
The main take-home messages of the present work can be summarized as follows: The main take-home messages of the present work can be summarized as follows:
\begin{itemize} \begin{itemize}
\item Within the SF-TD-DFT framework, we advice to use exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange. \item Within the SF-TD-DFT framework, we advice to use exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange.
This has been shown to be beneficial for the automerization barrier and the vertical excitation energies in the {\Dtwo} and {\Dfour} arrangements. This has been shown to be clearly beneficial for the automerization barrier and the vertical excitation energies computed at the {\Dtwo} and {\Dfour} equilibrium geometries.
\item At the SF-ADC level, we have found that the extended scheme, SF-ADC(2)-x, systematically worsen the results compared to the cheaper standard version, SF-ADC(2)-s. \item At the SF-ADC level, we have found that the extended scheme, SF-ADC(2)-x, systematically worsen the results compared to the cheaper standard version, SF-ADC(2)-s.
Moreover, as previously reported, SF-ADC(2)-s and SF-ADC(3) have opposite error patterns which means that SF-ADC(2.5) is an excellent compromise. Moreover, as previously reported, SF-ADC(2)-s and SF-ADC(3) have opposite error patterns which means that SF-ADC(2.5) is an excellent compromise.
@ -860,11 +860,11 @@ In such scenario, the SF-TD-DFT excitation energies can exhibit errors of the or
However, it was satisfying to see that the spin-flip version of ADC can lower these errors to \SIrange{0.1}{0.2}{\eV}. However, it was satisfying to see that the spin-flip version of ADC can lower these errors to \SIrange{0.1}{0.2}{\eV}.
\item Concerning the multi-configurational methods, we have found that NEVPT2 and CASPT2 can provide different excitation energies for the small (4e,4o) active space, but they becomes very similar when the larger (12e,12o) active space is considered. \item Concerning the multi-configurational methods, we have found that NEVPT2 and CASPT2 can provide different excitation energies for the small (4e,4o) active space, but they becomes very similar when the larger (12e,12o) active space is considered.
From a more general perspective, a significant difference between NEVPT2 and CASPT2 can be seen as a warning that the active space has been poorly chosen. From a more general perspective, a significant difference between NEVPT2 and CASPT2 is usually not a good sign and can be seen as a clear warning that the active space is too small or has been poorly chosen.
\item The ionic states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space. \item The ionic states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space.
\item In the context of CC methods, although the inclusion of triple excitations is very satisfactory in most cases, the inclusion of quadruples excitation (via CC4 or CCSDTQ) is mandatory to reach high accuracy (especially in the case of doubly-excited states). \item In the context of CC methods, although the inclusion of triple excitations (via CC3 or CCSDT) is very satisfactory in most cases, the inclusion of quadruples excitation (via CC4 or CCSDTQ) is mandatory to reach high accuracy (especially in the case of doubly-excited states).
We also point out that, within the error bar linked to the CIPSI extrapolation procedure, CCSDTQ and CIPSI yield similar excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states. We also point out that, within the error bar linked to the CIPSI extrapolation procedure, CCSDTQ and CIPSI yield similar excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states.
\end{itemize} \end{itemize}