saving work in D4h
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Pierre-Francois Loos
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@ -112,7 +112,7 @@ 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, {\sBoneg}, of the square arrangement is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}), while the lowest triplet state, {\Atwog}, is a minimum on the triplet potential energy surface.
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Thus, the automerization 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 \SIrange{1.6}{10}{\kcalmol}, \cite{Whitman_1982} while previous multi-reference calculations yield an energy barrier in the range of \SIrange{6}{7}{\kcalmol}. \cite{Eckert-Maksic_2006}
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The energy of this barrier is estimated, experimentally, in the range of \SIrange{1.6}{10}{\kcalmol}, \cite{Whitman_1982} while previous state-of-the-art \textit{ab initio} calculations yield an energy barrier in the range of \SIrange{7}{9}{\kcalmol}. \cite{Eckert-Maksic_2006,Li_2009,Shen_2012,Zhang_2019}
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The lowest-energy excited states of CBD in both symmetries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and the {\sBoneg} and {\Atwog} states for the square one.
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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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@ -263,7 +263,7 @@ If neither CC4, nor CCSDT are feasible, then we rely on NEVPT2(12,12).
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The procedure for each extrapolated value is explicitly mentioned as a footnote.
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Note that, due to error bar inherently linked to the CIPSI calculations (see Subsection \ref{sec:SCI}), these are mostly used as an additional safety net to further check the convergence of the CCSDTQ estimates.
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A table gathering these TBEs as well as literature data for the automerization barrier and the vertical excitation energies can be found in the {\SupInf}.
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Tables gathering these TBEs as well as literature data for the automerization barrier and the vertical excitation energies can be found in the {\SupInf}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -375,7 +375,8 @@ CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[ 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\
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%%% FIGURE II %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{AB_AVTZ}
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\caption{Automerization barrier (in \si{\kcalmol}) of CBD at various levels of theory using the aug-cc-pVTZ basis.}
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\caption{Automerization barrier (in \si{\kcalmol}) of CBD at various levels of theory using the aug-cc-pVTZ basis.
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See {\SupInf} for the raw data.}
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\label{fig:AB}
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\end{figure*}
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%%% %%% %%% %%%
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@ -422,7 +423,7 @@ Note that the introduction of the triple excitations is clearly mandatory to hav
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\begin{table}
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\caption{
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Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.}
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\label{tab:sf_tddft_D2h}
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\label{tab:sf_D2h}
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\begin{ruledtabular}
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\begin{tabular}{llrrr}
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& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
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@ -506,19 +507,6 @@ SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
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\cline{3-5}
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Method & Basis & {\tBoneg} & {\sBoneg} & {\twoAg} \\
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\hline
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CC3 &6-31+G(d)& $1.420$ & $3.341$ & $4.658$ \\
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& aug-cc-pVDZ & $1.396$ & $3.158$ & $4.711$ \\
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& aug-cc-pVTZ & $1.402$ & $3.119$ & $4.777$ \\
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& aug-cc-pVQZ & $1.409$ & $3.113$ & $4.774$ \\[0.1cm]
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CCSDT &6-31+G(d)& $1.442$ & $3.357$ & $4.311$ \\
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& aug-cc-pVDZ & $1.411$ & $3.175$ & $4.327$ \\
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& aug-cc-pVTZ & $1.411$ & $3.139$ & $4.429$ \\[0.1cm]
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CC4 &6-31+G(d)& & $3.343$ & $4.067$ \\
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& aug-cc-pVDZ & & $3.164$ & $4.041$ \\
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& aug-cc-pVTZ & & $\left[3.128\right]$\fnm[1] & $\left[4.143\right]$\fnm[1]\\[0.1cm]
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CCSDTQ &6-31+G(d)& & $3.340$ & $4.073$ \\
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& aug-cc-pVDZ & & $\left[3.161\right]$\fnm[2]& $\left[4.047\right]$\fnm[2] \\
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& aug-cc-pVTZ & & $\left[3.125\right]$\fnm[3]& $\left[4.149\right]$\fnm[3]\\[0.1cm]
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CASSCF(4,4) &6-31+G(d)& $1.662$ & $4.657$ & $4.439$ \\
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& aug-cc-pVDZ & $1.672$ & $4.563$ & $4.448$ \\
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& aug-cc-pVTZ & $1.670$ & $4.546$ & $4.441$ \\
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@ -564,6 +552,19 @@ PC-NEVPT2(12,12) &6-31+G(d)& $1.487$ & $3.296$ & $4.103$ \\
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& aug-cc-pVTZ & $1.462$ & $3.063$ & $4.056$ \\
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& aug-cc-pVQZ & $1.464$ & $3.043$ & $4.059$ \\[0.1cm]
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%MRCI(12,12) &6-31+G(d)& & & $4.125$ \\[0.1cm]
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CC3 &6-31+G(d)& $1.420$ & $3.341$ & $4.658$ \\
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& aug-cc-pVDZ & $1.396$ & $3.158$ & $4.711$ \\
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& aug-cc-pVTZ & $1.402$ & $3.119$ & $4.777$ \\
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& aug-cc-pVQZ & $1.409$ & $3.113$ & $4.774$ \\[0.1cm]
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CCSDT &6-31+G(d)& $1.442$ & $3.357$ & $4.311$ \\
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& aug-cc-pVDZ & $1.411$ & $3.175$ & $4.327$ \\
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& aug-cc-pVTZ & $1.411$ & $3.139$ & $4.429$ \\[0.1cm]
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CC4 &6-31+G(d)& & $3.343$ & $4.067$ \\
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& aug-cc-pVDZ & & $3.164$ & $4.041$ \\
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& aug-cc-pVTZ & & $\left[3.128\right]$\fnm[1] & $\left[4.143\right]$\fnm[1]\\[0.1cm]
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CCSDTQ &6-31+G(d)& & $3.340$ & $4.073$ \\
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& aug-cc-pVDZ & & $\left[3.161\right]$\fnm[2]& $\left[4.047\right]$\fnm[2] \\
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& aug-cc-pVTZ & & $\left[3.125\right]$\fnm[3]& $\left[4.149\right]$\fnm[3]\\[0.1cm]
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CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
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& aug-cc-pVDZ & $1.458\pm 0.009$ & $3.187\pm 0.035$ & $4.04\pm 0.04$ \\
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& aug-cc-pVTZ & $1.461\pm 0.030$ & $3.142\pm 0.035$ & $4.03\pm 0.09$ \\
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@ -579,13 +580,14 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
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%%% FIGURE III %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{D2h}
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\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.}
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\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.
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See {\SupInf} for the raw data.}
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%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
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\label{fig:D2h}
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\end{figure*}
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%%% %%% %%% %%%
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Table \ref{tab:sf_tddft_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
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Table \ref{tab:sf_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
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Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation energies with respect to the level of theory is illustrated in Fig.~\ref{fig:D2h}.
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At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in the {\tBoneg}, {\sBoneg}, and {\twoAg} are 99\%, 95\%, and 1\%, respectively.
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@ -594,7 +596,7 @@ Therefore, the two formers are dominated by single excitations, while the latter
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First, let us discuss basis set effects at the SF-TD-DFT level.
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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}
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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.
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Functionals with large amount 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}
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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}
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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.
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The triplet state, {\tBoneg}, is much better described with errors below \SI{0.1}{\eV}.
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Note that, as evidenced by the data reported in {\SupInf}, none of these states exhibit a strong spin contamination.
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@ -629,40 +631,7 @@ Using a larger active resolves most of these issues: CASSCF predicts the correct
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Finally, for the CC models, 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.
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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.
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The CCSDTQ excitation energies (which are used to define the TBEs) are systematically within the error bar of the CIPSI calculations.
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%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}.
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%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.
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%The {\sBoneg} state values in CASSCF(4,4) is much higher than for any of the other methods discussed so far.
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%With CASPT2(4,4) we retrieve the right ordering between the states and we see large energy differences with the CASSCF values.
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%Indeed, we have approximatively \SIrange{0.22}{0.25}{\eV} of energy difference for the triplet state for all bases and \SIrange{0.32}{0.36}{\eV} for the {\twoAg} state, the largest energy difference is for the {\sBoneg} state with \SIrange{1.5}{1.6}{\eV}.
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%For the XMS-CASPT2(4,4) only the {\twoAg} state is described with values similar than for the CAPST2(4,4) method.
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%For the NEVPT2(4,4) methods (SC-NEVPT2 and PC-NEVPT2) the vertical energies are similar for the {\sBoneg} and the {\twoAg} states with approximatively \SIrange{0.002}{0.003}{\eV} and \SIrange{0.02}{0.03}{\eV} of energy difference for all bases, respectively.
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%The energy difference for the {\sBoneg} state is slightly larger with \SI{0.05}{\eV} for all bases.
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%Note that for this state the vertical energy varies of \SI{0.23}{eV} from the 6-31+G(d) basis to the aug-cc-pVDZ one.
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%Then we use a larger active space with twelve electrons in twelve orbitals, the CASSF(12,12) values are close to the CASSCF(4,4) value for the triplet state with 0.01-0.02 eV of energy differences.
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%For the {\twoAg} state we have an energy difference of about \SI{0.2}{eV} between the CASSCF(4,4) and the CASSCF(12,12) values.
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%We can notice that increasing the size of the active space gives the right ordering for the states and we have an energy difference of around \SI{0.7}{\eV} for the {\sBoneg} state between CASSCF(4,4) and the CASSCF(12,12) values.
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%The CASPT2(12,12) method decreases the energy of all states compared to the CASSCF(12,12) method, again the decrease is not the same for all states.
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%We have a diminution from CASSCF to CASPT2 of about \SIrange{0.17}{0.2}{\eV} for the {\tBoneg} and the {\twoAg} states and for the different bases.
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%Again, the energy difference for the {\twoAg} state is larger with \SIrange{0.5}{0.7}{\eV} depending on the basis.
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%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.
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%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.
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%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.
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%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.
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%The energy difference is larger for the {\twoAg} state with around \SIrange{0.35}{0.38}{\eV}.
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%The same observation can be done for CCSDTQ and CC4 with similar vertical energies for all bases.
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%Note that the {\tBoneg} state can not be described with these methods.
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%For the CC methods taking into account the quadruple excitations is necessary to have a good description of the doubly excited state {\twoAg}.
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%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}.
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%We see that the SF-TD-DFT vertical energies are rather close to the TBE one for the doubly excited state {\twoAg}.
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%Indeed, as presented later in subsection \ref{sec:TBE}, SF methods are able to describe effectively states with strong multi-configurational character.
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%The same observation can be done for the SF-ADC values but with much better results for the two other states.
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%Again we can notice that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the values.
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%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.
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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.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{{\Dfour} symmetry}
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@ -750,22 +719,6 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
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\cline{3-5}
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Method & Basis & {\Atwog} & {\Aoneg} & {\Btwog} \\
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\hline
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CCSD & 6-31+G(d) & $0.148$ & $1.788$ & \\
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& aug-cc-pVDZ & $0.100$ & $1.650$ & \\
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& aug-cc-pVTZ & $0.085$ & $1.600$ & \\
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& aug-cc-pVQZ & $0.084$ & $1.588$ & \\[0.1cm]
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CC3 & 6-31+G(d) & & $1.809$ & $2.836$ \\
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& aug-cc-pVDZ & & $1.695$ & $2.646$ \\
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& aug-cc-pVTZ & & $1.662$ & $2.720$ \\[0.1cm]
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CCSDT & 6-31+G(d) & $0.210$ & $1.751$ & $2.565$ \\
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& aug-cc-pVDZ & $0.165$ & $1.659$ & $2.450$ \\
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& aug-cc-pVTZ & $0.149$ & $1.631$ & $2.537$ \\[0.1cm]
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CC4 & 6-31+G(d) & & $1.604$ & $2.121$ \\
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& aug-cc-pVDZ & & $1.539$ & $1.934$ \\
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& aug-cc-pVTZ & & $\left[1.511 \right]$\fnm[1] &$\left[2.021 \right]$\fnm[1] \\[0.1cm]
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CCSDTQ & 6-31+G(d) & $0.205$ & $1.593$ & $2.134$ \\
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& aug-cc-pVDZ & $\left[0.160\right]$\fnm[2] & $\left[1.528 \right]$\fnm[4]&$\left[1.947\right]$\fnm[4] \\
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& aug-cc-pVTZ & $\left[0.144\right]$\fnm[3] & $\left[1.500 \right]$\fnm[5]&$\left[2.034\right]$\fnm[5] \\ [0.1cm]
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CASSCF(4,4) & 6-31+G(d) & $0.447$ & $2.257$ & $3.549$ \\
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& aug-cc-pVDZ & $0.438$ & $2.240$ & $3.443$ \\
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& aug-cc-pVTZ & $0.434$ & $2.234$ & $3.424$ \\
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@ -802,6 +755,22 @@ PC-NEVPT2(12,12) & 6-31+G(d) & $0.189$ & $1.579$ & $2.020$ \\
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& aug-cc-pVDZ & $0.156$ & $1.530$ & $1.854$ \\
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& aug-cc-pVTZ & $0.131$ & $1.476$ & $1.756$ \\
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& aug-cc-pVQZ & $0.126$ & $1.460$ & $1.727$ \\[0.1cm]
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CCSD & 6-31+G(d) & $0.148$ & $1.788$ & \\
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& aug-cc-pVDZ & $0.100$ & $1.650$ & \\
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& aug-cc-pVTZ & $0.085$ & $1.600$ & \\
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& aug-cc-pVQZ & $0.084$ & $1.588$ & \\[0.1cm]
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CC3 & 6-31+G(d) & & $1.809$ & $2.836$ \\
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& aug-cc-pVDZ & & $1.695$ & $2.646$ \\
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& aug-cc-pVTZ & & $1.662$ & $2.720$ \\[0.1cm]
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CCSDT & 6-31+G(d) & $0.210$ & $1.751$ & $2.565$ \\
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& aug-cc-pVDZ & $0.165$ & $1.659$ & $2.450$ \\
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& aug-cc-pVTZ & $0.149$ & $1.631$ & $2.537$ \\[0.1cm]
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CC4 & 6-31+G(d) & & $1.604$ & $2.121$ \\
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& aug-cc-pVDZ & & $1.539$ & $1.934$ \\
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& aug-cc-pVTZ & & $\left[1.511 \right]$\fnm[1] &$\left[2.021 \right]$\fnm[1] \\[0.1cm]
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CCSDTQ & 6-31+G(d) & $0.205$ & $1.593$ & $2.134$ \\
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& aug-cc-pVDZ & $\left[0.160\right]$\fnm[2] & $\left[1.528 \right]$\fnm[4]&$\left[1.947\right]$\fnm[4] \\
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& aug-cc-pVTZ & $\left[0.144\right]$\fnm[3] & $\left[1.500 \right]$\fnm[5]&$\left[2.034\right]$\fnm[5] \\ [0.1cm]
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CIPSI & 6-31+G(d) & $0.201\pm 0.003$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
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& aug-cc-pVDZ & $0.157\pm 0.003$ & $1.587\pm 0.005$ & $2.102\pm 0.027$ \\
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& aug-cc-pVTZ & $0.169\pm 0.029$ & $1.63\pm 0.05$ & \\
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@ -820,14 +789,17 @@ CIPSI & 6-31+G(d) & $0.201\pm 0.003$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
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%%% FIGURE IV %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{D4h}
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\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.}
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\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.
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See {\SupInf} for the raw data.}
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%Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multi-reference, CC, and TBE values, respectively.}
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\label{fig:D4h}
|
||||
\end{figure*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
\alert{In the {\Dtwo} symmetry we could not compute $\%T_1$ values associated with the {\Atwog}, {\Aoneg}, and {\Btwog} excited states.
|
||||
However, it is clear from the inspection of the wave function that}
|
||||
In Table \ref{tab:sf_D4h}, we report, at the {\Dfour} square planar equilibrium geometry of the {\Atwog} state, the vertical transition energies associated with the {\Atwog}, {\Aoneg}, and {\Btwog} states obtained using the spin-flip formalism, while Table \ref{tab:D4h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
|
||||
The vertical excitation energies computed at various levels of theory are depicted in Fig.~\ref{fig:D4h} for the aug-cc-pVTZ basis.
|
||||
Unfortunately, due to technical limitations, we could not compute $\%T_1$ values associated with the {\Atwog}, {\Aoneg}, and {\Btwog} excited states in the {\Dfour} symmetry.
|
||||
However, it is clear from the inspection of the wave function that, with respect to the \sBoneg ground state, {\Atwog} and {\Btwog} are dominated by single excitations, while {\Aoneg} is strongly dominated by double excitations.
|
||||
|
||||
Table \ref{tab:sf_D4h} shows vertical energies obtained with spin-flip methods and Table \ref{tab:D4h} vertical energies obtained with standard methods.
|
||||
As for the previous geometry we start with the SF-TD-DFT results with hybrid functionals.
|
||||
@ -874,8 +846,6 @@ The energy difference is larger for the {\Btwog} state with about \SIrange{0.27}
|
||||
Next we have the NEVPT2 methods, the first observation that we can make is that by increasing the size of the active space we retrieve the right ordering of the {\Aoneg} and {\Btwog} states.
|
||||
Again, every vertical energy is higher in the NEVPT2(12,12) case than for the NEVPT2(4,4) one.
|
||||
|
||||
|
||||
|
||||
Figure \ref{fig:D4h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_D4h} and \ref{tab:D4h}.
|
||||
We see that all methods exhibit larger variations of the vertical energies due to the high symmetry of the {\Dfour} structure.
|
||||
For the triplet state most of the methods used are able to give a vertical energy close to the TBE one, we can although see that CASSCF, with the two different active spaces, show larger energy error than most of the DFT functionals used.
|
||||
@ -885,135 +855,6 @@ Multi-configurational methods with the smallest active space do not demonstrate
|
||||
Note that CASPT2 improve a lot the description of all the states compared to CASSCF.
|
||||
The various TD-DFT functionals are not able to describe correctly the two singlet excited states.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\subsubsection{Theoretical best estimates}
|
||||
%\label{sec:TBE}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
|
||||
|
||||
%================================================
|
||||
|
||||
%Table \ref{tab:TBE} shows the energy differences, for the automerization barrier (AB) and the different states considered, between the various methods treated and the Theoretical Best Estimate (TBE) at the aug-cc-pVTZ level for the AB and the states.
|
||||
%The percentage $\%T_1$ shown in parentheses for the excited states of the {\Dtwo} geometry is a metric that gives the percentage of single excitation calculated at the CC3/aug-cc-pVTZ level and it allows us to characterize the transition.
|
||||
%First, we look at the AB energy difference.
|
||||
%SF-TD-DFT shows large variations of the energy with errors of \SIrange{1.42}{10.81}{\kcalmol} compared to the TBE value.
|
||||
%SF-ADC schemes provide smaller errors with \SIrange{0.30}{1.44}{\kcalmol} where we have that the SF-ADC(2)-x gives a worse error than the SF-ADC(2)-s method.
|
||||
%CC methods also give small energy differences with \SIrange{0.11}{1.05}{\kcalmol} and where the CC4 provides an energy very close to the TBE one.
|
||||
%
|
||||
%Then we look at the vertical energy errors for the {\Dtwo} structure.
|
||||
%First we consider the {\tBoneg} state and we look at the SF-TD-DFT results.
|
||||
%We see that increasing the amount of exact exchange in the functional give closer results to the TBE, indeed we have \SI{0.24}{\eV} and \SI{0.22}{\eV} of errors for the B3LYP and the PBE0 functionals, respectively whereas we have an error of \SI{0.08}{\eV} for the BH\&HLYP functional.
|
||||
%For the other functionals we have errors of \SIrange{0.10}{0.43}{\eV}, note that for this state the M06-2X functional gives the same result than the TBE.
|
||||
%We can also notice that all the functionals considered overestimate the vertical energies.
|
||||
%The ADC schemes give closer energies with errors of \SIrange{0.04}{0.08}{\eV}, note that ADC(2)-x does not improve the result compared to ADC(2)-s and that ADC(3) understimate the vertical energy whereas ADC(2)-s and ADC(2)-x overestimate the vertical energy.
|
||||
%The CC3 and CCSDT results provide energy errors of \SIrange{0.05}{0.06}{\eV} respectively.
|
||||
%Then we go through the multi-reference methods with the two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals.
|
||||
%For the smaller active space we have errors of \SIrange{0.05}{0.21}{\eV}, the largest error comes from CASSCF(4,4) which is improved by CASPT2(4,4) that gives the smaller error.
|
||||
%Then for the largest active space multi-reference methods provide energy errors of \SIrange{0.02}{0.22}{\eV} with again the largest error coming from CASSCF(12,12) which is again improved by CASPT2(12,12) gives the smaller error.
|
||||
%
|
||||
%For the {\sBoneg} state of the {\Dtwo} structure we see that all the xc-functional underestimate the vertical excitation energy with energy differences of about \SIrange{0.35}{0.93}{\eV}.
|
||||
%The ADC values are much closer to the TBE with energy differences around \SIrange{0.03}{0.09}{\eV}.
|
||||
%Obviously, the CC vertical energies are close to the TBE one with around or less than \SI{0.01}{\eV} of energy difference.
|
||||
%For the CASSCF(4,4) vertical energy we have a large difference of around \SI{1.42}{\eV} compared to the TBE value due to the lack of dynamical correlation in the CASSCF method.
|
||||
%As previously seen the CAPT2(4,4) method correct this and we obtain a value of \SI{0.20}{\eV}.
|
||||
%The others multi-reference methods in this active space give energy differences of around \SIrange{0.55}{0.76}{\eV} compared the the TBE reference.
|
||||
%For the largest active space with twelve electrons in twelve orbitals we have an improvement of the vertical energies with \SI{0.72}{eV} of energy difference for the CASSCF(12,12) method and around 0.06 eV for the others multi-configurational methods.
|
||||
%
|
||||
%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 multi-configurational 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 multi-configurational character of the {\twoAg} state whereas the {\sBoneg} state has a very weak multi-configurational 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 multi-reference methods obviously give better results too for the {\twoAg} state.
|
||||
%
|
||||
%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 multi-reference 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 multi-reference 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 multi-reference 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 multi-reference 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 multi-configurational 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 multi-configurational character
|
||||
|
||||
|
||||
%%% TABLE I %%%
|
||||
%\begin{squeezetable}
|
||||
%\begin{table*}
|
||||
% \caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the automerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage $\%T_1$ calculated at the CC3/aug-cc-pVTZ level.}
|
||||
%
|
||||
% \label{tab:TBE}
|
||||
% \begin{ruledtabular}
|
||||
% \begin{tabular}{lrrrrrrr}
|
||||
%%\begin{tabular}{*{1}{*{8}{l}}}
|
||||
%&\mc{3}{r}{\Dtwo excitation energies (eV)} & \mc{3}{r}{\Dfour excitation energies (eV)} \\
|
||||
% \cline{3-5} \cline{6-8}
|
||||
%Method & AB & $1\,{}^3B_{1g} $~(98.7 \%) & $1\,{}^1B_{1g} $ (95.0 \%)& $2\,{}^1A_{g} $(0.84 \%) & $1\,{}^3A_{2g} $ & $2\,{}^1A_{1g} $ & $1\,{}^1B_{2g} $ \\
|
||||
% \hline
|
||||
%SF-TD-B3LYP & $10.41$ & $0.241$ & $-0.926$ & $-0.161$ & $-0.164$ & $-1.028$ & $-1.501$ \\
|
||||
%SF-TD-PBE0 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\
|
||||
%SF-TD-BHHLYP & $8.95$ & $0.078$ & $-0.393$ & $0.343$ & $-0.099$ & $-0.251$ & $-0.603$ \\
|
||||
%SF-TD-M06-2X & $1.42$ & $0.000$ & $-0.354$ & $0.208$ & $-0.066$ & $-0.097$ & $-0.432$ \\
|
||||
%SF-TD-CAM-B3LYP & $9.90$ & $0.280$ & $-0.807$ & $-0.011$ & $-0.134$ & $-0.920$ & $-1.370$ \\
|
||||
%SF-TD-$\omega $B97X-V & $10.01$ & $0.335$ & $-0.774$ & $0.064$ & $-0.118$ & $-0.928$ & $-1.372$ \\
|
||||
%SF-TD-M11 & $2.29$ & $0.097$ & $-0.474$ & $0.151$ & $-0.063$ & $-0.312$ & $-0.675$ \\
|
||||
%SF-TD-LC-$\omega $PBE08 & $10.81$ & $0.435$ & $-0.710$ & $0.199$ & $-0.086$ & $-0.939$ & $-1.376$ \\[0.1cm]
|
||||
%SF-ADC(2)-s & $-0.30$ & $0.069$ & $-0.026$ & $-0.018$ & $0.112$ & $0.112$ & $-0.190$ \\
|
||||
%SF-ADC(2)-x & $1.44$ & $0.077$ & $-0.094$ & $-0.446$ & $0.068$ & $-0.409$ & $-0.303$ \\
|
||||
%SF-ADC(3) & $0.65$ & $-0.043$ & $0.037$ & $0.075$ & $-0.065$ & $0.075$ & $-0.181$ \\[0.1cm]
|
||||
%CCSD & $0.95$ & & & & $-0.059$ & $0.100$ & \\
|
||||
%CC3 & $-1.05$ & $-0.060$ & $-0.006$ & $0.628$ & & $0.162$ & $0.686$ \\
|
||||
%CCSDT & $-0.25$ & $-0.051$ & $0.014$ & $0.280$ & $0.005$ & $0.131$ & $0.503$ \\
|
||||
%CC4 & $-0.11$ & & $0.003$ & $-0.006$ & & $0.011$ & $-0.013$ \\
|
||||
%CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
|
||||
%CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
|
||||
%CASPT2(4,4) & & $-0.050$ & $-0.202$ & $-0.077$ & $-0.016$ & $0.006$ & $-0.399$ \\
|
||||
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
|
||||
%SC-NEVPT2(4,4) & & $-0.083$ & $-0.703$ & $-0.041$ & $-0.120$ & $-0.072$ & $-0.979$ \\
|
||||
%PC-NEVPT2(4,4) & & $-0.080$ & $-0.757$ & $-0.066$ & $-0.118$ & $-0.097$ & $-1.031$ \\
|
||||
%MRCI(4,4) & & $0.106$ & $0.553$ & $0.121$ & $0.127$ & $0.324$ & $0.381$ \\[0.1cm]
|
||||
%CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
|
||||
%CASPT2(12,12) & & $0.018$ & $0.058$ & $-0.106$ & $0.039$ & $0.038$ & $-0.108$ \\
|
||||
%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
|
||||
%SC-NEVPT2(12,12) & $-0.65$ & $0.039$ & $0.063$ & $-0.063$ & $0.021$ & $0.046$ & $-0.142$ \\
|
||||
%PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.278$ \\[0.1cm]
|
||||
%%CIPSI & & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ & \\
|
||||
%\bf{TBE} & $\left[\bf{8.93}\right]$\fnm[1] & $\left[\bf{1.462}\right]$\fnm[2] & $\left[\bf{3.125}\right]$\fnm[3] & $\left[\bf{4.149}\right]$\fnm[3] & $\left[\bf{0.144}\right]$\fnm[4] & $\left[\bf{1.500}\right]$\fnm[3] & $\left[\bf{2.034}\right]$\fnm[3] \\
|
||||
%\end{tabular}
|
||||
%
|
||||
% \end{ruledtabular}
|
||||
% \fnt[1]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.}
|
||||
% \fnt[2]{Value obtained using the NEVPT2(12,12) one.}
|
||||
% \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.}
|
||||
%
|
||||
%\end{table*}
|
||||
%\end{squeezetable}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Conclusion}
|
||||
\label{sec:conclusion}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user