D4h and D2h figures

Manuscript/CBD.tex

@@ -276,6 +276,25 @@ In both structures the CBD has a singlet ground state, for the spin-flip calcula
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Theoretical Best Estimate (TBE)}
+For each studied quantity, i.e., the autoisomerisation barrier and the vertical excitations, we provide a theoretical best estimate (TBE). These TBEs are provided using extrapolated CCSDTQ/AVTZ values when possible and using NEVPT2(12,12) otherwise. The extrapolation of the CCSDTQ/AVTZ values is done using two schemes. The first one uses CC4 values for the extrapolation and proceed as follows
+
+\begin{equation}
+\Delta E^{\text{CCSDTQ}}_{\text{AVTZ}} = \Delta E^{\text{CCSDTQ}}_{\text{AVDZ}} + \left[ \Delta E^{\text{CC4}}_{\text{AVTZ}} - \Delta E^{\text{CC4}}_{\text{AVDZ}} \right]
+\end{equation}
+
+and we evaluate the CCSDTQ/AVTZ values as 
+
+\begin{equation}
+\Delta  E^{\text{CCSDTQ}}_{\text{AVDZ}} =  \Delta E^{\text{CCSDTQ}}_{6-31\text{G}+\text{(d)}} +  \left[ \Delta E^{\text{CC4}}_{\text{AVDZ}} - \Delta E^{\text{CC4}}_{6-31\text{G}+\text{(d)}} \right]
+\end{equation}
+when CC4/AVTZ values have been obtained. If it is not the case we extrapolate CC4/AVTZ values using the CCSDT ones as follows
+
+\begin{equation}
+\Delta E^{\text{CC4}}_{\text{AVTZ}} = \Delta E^{\text{CC4}}_{\text{AVDZ}} + \left[ \Delta E^{\text{CCSDT}}_{\text{AVTZ}} - \Delta E^{\text{CCSDT}}_{\text{AVDZ}} \right]
+\end{equation}
+Then if the CC4 values have not been obtained then we use the second scheme which is the same as the first one but instead of the CC4 values we use CCSDT to extrapolate CCSDTQ. If none of the two schemes is possible then we use the NEVPT2(12,12) values. Note that a NEVPT2(12,12) value is used only once for one vertical excitation of the $D_{4h}$ structure.
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -380,7 +399,7 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ &  &  \\
 
 \begin{figure*}
 \includegraphics[scale=0.5]{AB_AVTZ.pdf}
-\caption{Here comes the caption of the figure}
+\caption{Autoisomerization barrier energy for the CBD molecule using the AVTZ basis. Purple histograms are for the SF-TD-DFT functionals, orange histograms are for the SF-ADC schemes, green histograms are for the multireference methods, blue histograms are for the CC methods and the black one is for the TBE.}
 \end{figure*}
 
 %%% %%% %%% %%%
@@ -712,25 +731,33 @@ CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
 \end{squeezetable}
 %%% %%% %%% %%%
 
+\subsubsection{TBE}
+
 Table \ref{tab:TBE} shows the energy differences, for the autoisomerization barrier (AB) and the different states considered, between the various methods treated and the Theoretical Best Estimate (TBE) at the AVTZ level for the AB and the states.  First, we look at the AB energy difference. SF-TD-DFT shows large variations of the energy with errors of 1.42-10.81 \kcalmol compared to the TBE value. SF-ADC schemes provide smaller errors with 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 0.11-1.05 \kcalmol and where the CC4 provides an energy very close to the TBE one.
 
-For the $1\,{}^1B_{1g} $ state of the $(D_{2h})$ structure we see that all the xc-functional underestimate the vertical excitation energy with energy differences of about 0.35-0.93 eV. The ADC values are much closer to the TBE with energy differences around 0.03-0.09 eV. Obviously, the CC vertical energies are close to the TBE one with around or less than 0.01 eV of energy difference. For the CASSCF(4,4) vertical energy we have a large difference of around 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 0.20 eV. The others multireference methods in this active space give energy differences of around 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 0.72 eV of energy difference for the CASSCF(12,12) method and around 0.06 eV for the others multiconfigurational methods. The CIPSI method gives a vertical energy close to the TBE one with around 0.02 eV of energy difference. 
+Then we look at the vertical energy errors for the $(D_{2h})$ structure. First we consider the $1\,{}^3B_{1g} $ 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 0.24 and 0.22 eV of errors for the B3LYP and the PBE0 functionals, respectively whereas we have an error of 0.08 eV for the BH\&HLYP functional. For the other functionals we have errors of 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 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 0.05-0.06 eV respectively. Then we go through the multireference 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 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 multireference methods provide energy errors of 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 $1\,{}^1B_{1g} $ state of the $(D_{2h})$ structure we see that all the xc-functional underestimate the vertical excitation energy with energy differences of about 0.35-0.93 eV. The ADC values are much closer to the TBE with energy differences around 0.03-0.09 eV. Obviously, the CC vertical energies are close to the TBE one with around or less than 0.01 eV of energy difference. For the CASSCF(4,4) vertical energy we have a large difference of around 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 0.20 eV. The others multireference methods in this active space give energy differences of around 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 0.72 eV of energy difference for the CASSCF(12,12) method and around 0.06 eV for the others multiconfigurational methods. 
+
+Then, for the $2\,{}^1A_{g} $ state we obtain closer results of the SF-TD-DFT methods to the TBE than in the case of the $1\,{}^1B_{1g} $ state. Indeed, we have an energy difference of about 0.01-0.34 eV for the $2\,{}^1A_{g} $ state whereas we have 0.35-0.93 eV for the $1\,{}^1B_{1g} $ state. The ADC schemes give the same error to the TBE value than for the other singlet state with 0.02 eV for the ADC(2) scheme and 0.07 eV for the ADC(3) one. The ADC(2)-x scheme provides a larger error with 0.45 eV of energy difference. Here, the CC methods manifest more variations with 0.63 eV for the CC3 value and 0.28 eV for the CCSDT compared to the TBE values. The CC4 method provides a small error with less than 0.01 eV of energy difference. The multiconfigurational methods globally give smaller error than for the other singlet state with, for the two active spaces, 0.03-0.12 eV compared to the TBE value. We can notice that CC3 and CCSDT provide larger energy errors for the $2\,{}^1A_{g} $ state than for the $1\,{}^1B_{1g} $ state, this is due to the strong multiconfigurational character of the $2\,{}^1A_{g} $ state whereas the $1\,{}^1B_{1g} $ state has a very weak multiconfigurational character. It is interesting to see that SF-TD-DFT and SF-ADC methods give better results compared to the TBE than CC3 and even CCSDT meaning that spin-flip methods are able to describe double excited states. Note that multireference methods obviously give better results too for the $2\,{}^1A_{g} $ state.
+
+Finally we look at the vertical energy errors for the $(D_{4h})$ structure. First, we consider the $1\,{}^3A_{2g} $ state, the SF-TD-DFT methods give errors of about 0.07-1.6 eV where the largest energy differences are provided by the Hybrid functionals. The ADC schemes give similar error with around 0.06-1.1 eV of energy difference. For the CC methods we have an energy error of 0.06 eV for CCSD and less than 0.01 eV for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) 0.29 eV of error and 0.02 eV for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about 0.12-0.13 eV. A larger active space shows again an improvement with 0.23 eV of error for CASSCF(12,12) and around 0.01-0.04 eV for the other multireference methods. CIPSI provides similar error with 0.02 eV. Then, we look at the $2\,{}^1A_{1g}$ state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about 0.10-1.03 eV. The ADC schemes give better errors with around 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 0.10-0.16 eV and CC4 provides really close energy to the TBE one with 0.01 eV of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of 0.01-0.73 eV and 0.02-0.44 eV respectively with the largest errors coming from the CASSCF method. Lastly, we look at the $1\,{}^1B_{2g}$ state where we have globally larger errors. The SF-TD-DFT exhibits errors of 0.43-1.50 eV whereas ADC schemes give errors of 0.18-0.30 eV. CC3 and CCSDT provide energy differences of 0.50-0.69 eV and the CC4 shows again close energy to the CCSDTQ TBE energy with 0.01 eV of error. The multireference methods give energy differences of 0.38-1.39 eV and 0.11-0.60 eV for the four by four and twelve by twelve active space respectively. Here again we can make the same comment for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states. The first state has a strong multiconfigurational character but shows small errors through all methods and the latter has a weak multiconfigurational character but larger errors
 
-Then, for the $2\,{}^1A_{g} $ state we obtain closer results of the SF-TD-DFT methods to the TBE than in the case of the $1\,{}^1B_{1g} $ state. Indeed, we have an energy difference of about 0.01-0.34 eV for the $2\,{}^1A_{g} $ state whereas we have 0.35-0.93 eV for the $1\,{}^1B_{1g} $ state. The ADC schemes give the same error to the TBE value than for the other singlet state with 0.02 eV for the ADC(2) scheme and 0.07 eV for the ADC(3) one. The ADC(2)-x scheme provides a larger error with 0.45 eV of energy difference. Here, the CC methods manifest more variations with 0.63 eV for the CC3 value and 0.28 eV for the CCSDT compared to the TBE values. The CC4 method provides a small error with less than 0.01 eV of energy difference. The multiconfigurational methods globally give smaller error than for the other singlet state with, for the two acitve spaces, 0.03-0.12 eV compared to the TBE value. CIPSI displays a larger error than for the previous state with 0.12 eV. The results obtained for the $2\,{}^1A_{g} $ state can be surprising due to the strong multiconfigurational character of this state compared to the $1\,{}^1B_{1g} $ state which has a very weak multiconfigurational character. This can be explained by the fact that the TBE value which is obtained at the CCSDTQ level does not provide a good description of this state. 
 
-Finally we look at the vertical energy errors for the $(D_{4h})$ structure. First, we consider the $1\,{}^3A_{2g} $ state, the SF-TD-DFT methods give errors of about 0.07-1.6 eV where the largest energy differences are provided by the Hybrid functionals. The ADC schemes give similar error with around 0.06-1.1 eV of energy difference. For the CC methods we have an energy error of 0.06 eV for CCSD and less than 0.01 eV for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) 0.29 eV of error and 0.02 eV for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about 0.12-0.13 eV. A larger active space shows again an improvement with 0.23 eV of error for CASSCF(12,12) and around 0.01-0.04 eV for the other multireference methods. CIPSI provides similar error with 0.02 eV. Then, we look at the $2\,{}^1A_{1g}$ state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about 0.10-1.03 eV. The ADC schemes give better errors with around 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 0.10-0.16 eV and CC4 provides really close energy to the TBE one with 0.01 eV of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of 0.01-0.73 eV and 0.02-0.44 eV respectively with the largest errors coming from the CASSCF method. CIPSI gives an error of 0.13 eV compared to the TBE. Lastly, we look at the $1\,{}^1B_{2g}$ state where we have globally larger errors. The SF-TD-DFT exhibits errors of 0.43-1.50 eV whereas ADC schemes give errors of 0.18-0.30 eV. CC3 and CCSDT provide energy differences of 0.50-0.69 eV and the CC4 shows again close energy to the CCSDTQ TBE energy with 0.01 eV of error. The multireference methods give energy differences of 0.38-1.39 eV and 0.11-0.60 eV for the four by four and twelve by twelve active space respectively. Here again we can make the same comment for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states where the first has a strong multiconfigurational character but shows small errors through all methods and the latter has a weak multiconfigurational character but larger erros due to the quality of the TBE reference.
 
 
 %%% TABLE I %%%
 \begin{squeezetable}
 \begin{table*}
-	\caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV.}
+	\caption{Energy differences between the various methods and the TBE considered. Note that AB stands for the autoisomerization barrier and that the energies are in \kcalmol while the vertical energies are given in eV. The number in parenthesis is the percentage T1 calculated at the CC3/AVTZ level.}
 	
 	\label{tab:TBE}
 	\begin{ruledtabular}
 	\begin{tabular}{lrrrrrrr}
 %\begin{tabular}{*{1}{*{8}{l}}}
-Method		&	AB 	&	$1\,{}^3B_{1g} (D_{2h})$	&	$1\,{}^1B_{1g} (D_{2h})$	&	$2\,{}^1A_{g} (D_{2h})$ & 	$1\,{}^3A_{2g} (D_{4h})$	&	$2\,{}^1A_{1g} (D_{4h})$	&	$1\,{}^1B_{2g} (D_{4h})$ \\
+&\mc{3}{r}{$D_{2h}$ excitation energies (eV)} & \mc{3}{r}{$D_{4h}$ excitation energies (eV)} \\
+									\cline{3-5}  \cline{6-8}
+Method		&	AB 	&	$1\,{}^3B_{1g} $	&	$1\,{}^1B_{1g} $	&	$2\,{}^1A_{g} $ & 	$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 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\
@@ -744,7 +771,7 @@ SF-ADC(2)-s & $-0.30$ & $0.069$ & $-0.026$ & $-0.018$ & $0.112$ & $0.112$ & $-0.
 SF-ADC(2)-x & $1.44$ & $0.077$ & $-0.094$ & $-0.446$ & $0.068$ & $-0.409$ & $-0.303$ \\
 SF-ADC(3) & $0.65$ & $-0.043$ & $0.037$ & $0.075$ & $-0.065$ & $0.075$ & $-0.181$ \\
 CCSD & $0.95$ &  &  &  & $-0.059$ & $0.100$ &  \\
-CC3 & $-1.05$ & $-0.060$ & $-0.006$ & $0.628$ &  & $0.162$ & $0.686$ \\
+CC3 & $-1.05$ & $-0.060$ (98.7 \%)  & $-0.006$ (95.0 \%)  & $0.628$ (0.84 \%) &  & $0.162$ & $0.686$ \\
 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 & $0.00$ &  & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\
@@ -759,7 +786,7 @@ 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) &  & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.278$ \\
-CIPSI &  & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ &  \\
+%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}
 
@@ -774,6 +801,19 @@ CIPSI &  & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm
 %%% %%% %%% %%%
 
 
+\begin{figure*}
+%width=0.8\linewidth
+	\includegraphics[scale=0.5]{D2h.pdf}
+	\caption{Vertical energies of the $1\,{}^3B_{1g} $, $1\,{}^1B_{1g} $	and $2\,{}^1A_{g} $ states for the $D_{2h}$ geometry using the AVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
+\label{fig:D2h}
+\end{figure*}
+
+\begin{figure*}
+%width=0.8\linewidth
+	\includegraphics[scale=0.5]{D4h.pdf}
+	\caption{Vertical energies of the $1\,{}^3A_{2g} $, $2\,{}^1A_{1g} $	and $1\,{}^1B_{2g} $ states for the $D_{2h}$ geometry using the AVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
+\label{fig:D4h}
+\end{figure*}
 
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