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@ -138,9 +138,9 @@ One can address part of this issue by increasing the excitation order or by comp
both solutions being associated with an additional computational cost. both solutions being associated with an additional computational cost.
In the present work, we investigate the accuracy of each family of computational methods mentioned above on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries. In the present work, we investigate the accuracy of each family of computational methods mentioned above on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
Computational details are reported in Section \ref{sec:compmet}. Computational details are reported in Sec.~\ref{sec:compmet}.
Section \ref{sec:res} is devoted to the discussion of our results. Section \ref{sec:res} is devoted to the discussion of our results.
Finally, our conclusions are drawn in Section \ref{sec:conclusion}. Finally, our conclusions are drawn in Sec.~\ref{sec:conclusion}.
%%% FIGURE 1 %%% %%% FIGURE 1 %%%
\begin{figure} \begin{figure}
@ -228,6 +228,8 @@ 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.
More importantly, for each studied quantity (i.e., the autoisomerisation barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis. More importantly, for each studied quantity (i.e., the autoisomerisation barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis.
@ -594,35 +596,33 @@ However, their overall accuracy remains average especially for the singlet state
The triplet state, {\tBoneg}, is much better described with errors below \SI{0.1}{\eV}. The triplet state, {\tBoneg}, is much better described with errors below \SI{0.1}{\eV}.
Note that, as evidenced by the data reported in {\SupInf}, none of these states exhibit a strong spin contamination. Note that, as evidenced by the data reported in {\SupInf}, none of these states exhibit a strong spin contamination.
First, we discuss the SF-TD-DFT values with hybrid functionals. %First, we discuss the SF-TD-DFT values with hybrid functionals.
For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the {\tBoneg} state with \SI{0.012}{\eV}. %For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the {\tBoneg} state with \SI{0.012}{\eV}.
The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also observe that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the {\sBoneg} and the {\tBoneg} states. %The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also observe that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the {\sBoneg} and the {\tBoneg} states.
Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the {\tBoneg} and the {\oneAg} states. %Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the {\tBoneg} and the {\oneAg} states.
We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the {\tBoneg} state from PBE0 to BH\&HLYP is around \SI{0.1}{\eV} whereas for the {\sBoneg} and the {\twoAg} states this energy variation is about \SIrange{0.4}{0.5}{\eV} and \SI{0.34}{eV} respectively. %We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the {\tBoneg} state from PBE0 to BH\&HLYP is around \SI{0.1}{\eV} whereas for the {\sBoneg} and the {\twoAg} states this energy variation is about \SIrange{0.4}{0.5}{\eV} and \SI{0.34}{eV} respectively.
For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the {\sBoneg} and the {\tBoneg} states is larger than for the $\omega$B97X-V and LC-$\omega$PBE08 functionals despite the fact that the latter ones have a larger amount of exact exchange. %For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the {\sBoneg} and the {\tBoneg} states is larger than for the $\omega$B97X-V and LC-$\omega$PBE08 functionals despite the fact that the latter ones have a larger amount of exact exchange.
However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional. %However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional.
The M06-2X functional is an hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around \SIrange{0.03}{0.08}{\eV}. %The M06-2X functional is an hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around \SIrange{0.03}{0.08}{\eV}.
We can notice that the upper bound of \SI{0.08}{\eV} in the energy differences is due to the {\tBoneg} state. %We can notice that the upper bound of \SI{0.08}{\eV} in the energy differences is due to the {\tBoneg} state.
The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with \SIrange{0.01}{0.02}{\eV} and \SIrange{0.08}{0.10}{\eV} of energy difference, respectively. %The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with \SIrange{0.01}{0.02}{\eV} and \SIrange{0.08}{0.10}{\eV} of energy difference, respectively.
For the {\twoAg} state the M11 energies are closer to the $\omega$B97X-V ones with \SIrange{0.05}{0.09}{\eV} of energy difference. %For the {\twoAg} state the M11 energies are closer to the $\omega$B97X-V ones with \SIrange{0.05}{0.09}{\eV} of energy difference.
Second, we discuss the various SF-ADC schemes, \ie, SF-ADC(2)-s, SF-ADC(2)-x, and SF-ADC(3).
At the SF-ADC(2)-s level, going from the smallest 6-31+G(d) basis to the largest aug-cc-pVQZ basis, we see a small decrease in vertical excitation energies of about \SI{0.03}{\eV} for the {\tBoneg} state and around \SI{0.06}{\eV} for {\twoAg} state, while the transition energy of the {\sBoneg} state drops more significantly by about \SI{0.2}{\eV}.
(The SF-ADC(2)-x and SF-ADC(3) calculations with aug-cc-pVQZ were not feasible with our computational resources.)
These basis set effects are fairly transferable to the other wave function methods that we have considered here.
This further motivates the ``pyramidal'' extrapolation scheme that we have employed to produce the TBE values (see Sec.~\ref{sec:TBE_compdet}).
Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the present context with much larger errors that the other schemes.
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}.
Then we discuss the various ADC scheme (ADC(2)-s, ADC(2)-x and ADC(3)) results. For ADC(2) we have vertical energy differences of about 0.03 eV for the {\tBoneg} state and around 0.06 eV for {\twoAg} state throughout all bases.
However for the {\sBoneg} state we have an energy difference of about 0.2 eV.
For the ADC(2)-x and ADC(3) schemes the calculations with the aug-cc-pVQZ basis are not feasible with our resources.
With ADC(2)-x we have similar vertical energies for the triplet and the {\sBoneg} states but for the {\twoAg} state the energy difference between the ADC(2) and ADC(2)-x schemes is about \SIrange{0.4}{0.5}{\eV}.
The ADC(3) values are closer from the ADC(2) than the ADC(2)-x except for the triplet state for which we have a energy difference of about \SIrange{0.09}{0.14}{\eV}.
Now, we look at Table \ref{tab:D2h} to discuss the results of standard methods.
First we discuss the CC values, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively.
We can notice that for the {\tBoneg} and the {\sBoneg} states the CCSDT and the CC3 values are close with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases.
The energy difference is larger for the {\twoAg} state with around \SIrange{0.35}{0.38}{\eV}.
The same observation can be done for CCSDTQ and CC4 with similar vertical energies for all bases.
Note that the {\tBoneg} state can not be described with these methods.
\alert{Due to the lack of dynamic correlation, CASSCF poorly describes the {\sBoneg} ionic state. \alert{Due to the lack of dynamic correlation, CASSCF poorly describes the {\sBoneg} ionic state.
Although the autoisomerization barrier is not bad with the (4e,4o) active space, the {\sBoneg} is poorly describe mainly due to the poor CASSCF treatement that the second-order correction cannot fix.} Although the autoisomerization barrier is not bad with the (4e,4o) active space, the {\sBoneg} is poorly describe mainly due to the poor CASSCF treatement that the second-order correction cannot fix.}
Let us now move to discussion of the results of standard wave function methods that are reported in Table \ref{tab:D2h}.
Then we review the vertical energies obtained with multi-reference methods. Then we review the vertical energies obtained with multi-reference methods.
The smallest active space considered is four electrons in four orbitals, for CASSCF(4,4) we have small energy variations throughout bases for the {\tBoneg} and the {\twoAg} states but a larger variation for the {\sBoneg} state with around \SI{0.1}{\eV}. For the multi-configurational methods, the smallest active space considered is four electrons in four orbitals, for CASSCF(4,4) we have small energy variations throughout bases for the {\tBoneg} and the {\twoAg} states but a larger variation for the {\sBoneg} state with around \SI{0.1}{\eV}.
We can observe that we have the inversion of the states compared to all methods discussed so far between the {\twoAg} and {\sBoneg} states with {\sBoneg} higher than {\twoAg} due to the lack of dynamical correlation in the CASSCF methods. We can observe that we have the inversion of the states compared to all methods discussed so far between the {\twoAg} and {\sBoneg} states with {\sBoneg} higher than {\twoAg} due to the lack of dynamical correlation in the CASSCF methods.
The {\sBoneg} state values in CASSCF(4,4) is much higher than for any of the other methods discussed so far. The {\sBoneg} state values in CASSCF(4,4) is much higher than for any of the other methods discussed so far.
With CASPT2(4,4) we retrieve the right ordering between the states and we see large energy differences with the CASSCF values. With CASPT2(4,4) we retrieve the right ordering between the states and we see large energy differences with the CASSCF values.
@ -640,14 +640,20 @@ Again, the energy difference for the {\twoAg} state is larger with \SIrange{0.5}
In a similar way than with XMS-CASPT2(4,4), XMS-CASPT(12,12) only describes the {\twoAg} state and the vertical energies for this state are close to the CASPT(12,12) values. In a similar way than with XMS-CASPT2(4,4), XMS-CASPT(12,12) only describes the {\twoAg} state and the vertical energies for this state are close to the CASPT(12,12) values.
For the NEVPT2(12,12) schemes we see that for the triplet and the {\twoAg} states the energies are similar with an energy difference between the SC-NEVPT2(12,12) and the PC-NEVPT2(12,12) values of about \SIrange{0.03}{0.04}{\eV} and \SIrange{0.02}{0.03}{\eV} respectively. For the NEVPT2(12,12) schemes we see that for the triplet and the {\twoAg} states the energies are similar with an energy difference between the SC-NEVPT2(12,12) and the PC-NEVPT2(12,12) values of about \SIrange{0.03}{0.04}{\eV} and \SIrange{0.02}{0.03}{\eV} respectively.
Figure \ref{fig:D2h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_tddft_D2h} and \ref{tab:D2h}. First we discuss the CC values, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively.
We see that the SF-TD-DFT vertical energies are rather close to the TBE one for the doubly excited state {\twoAg}. We can notice that for the {\tBoneg} and the {\sBoneg} states the CCSDT and the CC3 values are close with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases.
Indeed, as presented later in subsection \ref{sec:TBE}, SF methods are able to describe effectively states with strong multi-configurational character. The energy difference is larger for the {\twoAg} state with around \SIrange{0.35}{0.38}{\eV}.
The same observation can be done for the SF-ADC values but with much better results for the two other states. The same observation can be done for CCSDTQ and CC4 with similar vertical energies for all bases.
Again we can notice that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the values. Note that the {\tBoneg} state can not be described with these methods.
For the multi-reference methods we see that using the smallest active space do not provide a good description of the {\sBoneg} state and in the case of CASSCF(4,4), as previously said, we even have {\sBoneg} state above the {\twoAg} state.
For the CC methods taking into account the quadruple excitations is necessary to have a good description of the doubly excited state {\twoAg}. For the CC methods taking into account the quadruple excitations is necessary to have a good description of the doubly excited state {\twoAg}.
%Figure \ref{fig:D2h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_tddft_D2h} and \ref{tab:D2h}.
%We see that the SF-TD-DFT vertical energies are rather close to the TBE one for the doubly excited state {\twoAg}.
%Indeed, as presented later in subsection \ref{sec:TBE}, SF methods are able to describe effectively states with strong multi-configurational character.
%The same observation can be done for the SF-ADC values but with much better results for the two other states.
%Again we can notice that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the values.
%For the multi-reference methods we see that using the smallest active space do not provide a good description of the {\sBoneg} state and in the case of CASSCF(4,4), as previously said, we even have {\sBoneg} state above the {\twoAg} state.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{{\Dfour} symmetry} \subsubsection{{\Dfour} symmetry}
\label{sec:D4h} \label{sec:D4h}