almost done with AB

Manuscript/CBD.tex

@@ -139,9 +139,7 @@ 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. 
 Computational details are reported in Section \ref{sec:compmet}.
-% for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multi-configurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods. 
 Section \ref{sec:res} is devoted to the discussion of our results. 
-%First, we consider the autoisomerization barrier (Subsection \ref{sec:auto}) and then the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries. 
 Finally, our conclusions are drawn in Section \ref{sec:conclusion}.
 
 %%% FIGURE 1 %%%
@@ -157,7 +155,6 @@ Finally, our conclusions are drawn in Section \ref{sec:conclusion}.
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 \section{Computational details}
 \label{sec:compmet}
-%The system under investigation in this work is the cyclobutadiene (CBD) molecule, rectangular (\Dtwo) and square (\Dfour) geometries are considered.  The (\Dtwo) geometry is obtained at the CC3 level without frozen core using the aug-cc-pVTZ and the (\Dfour) geometry is obtained at the RO-CCSD(T) level using aug-cc-pVTZ again without frozen core. All the calculations are performed using four basis set, the 6-31+G(d) basis and the aug-cc-pVXZ with X$=$D, T, Q. In the following we will use the notation AVXZ for the aug-cc-pVXZ basis, again with X$=$D, T, Q. The $\%T_1$ metric that gives the percentage of single excitation calculated at the CC3 level in \textcolor{red}{DALTON} allows to characterize the various states.Throughout all this work, spin-flip and spin-conserved calculations are performed with a UHF reference. 
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@@ -189,8 +186,7 @@ In some cases, we have also computed (singlet and triplet) excitation energies a
 
 To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell or unrestricted formalisms, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state {\Aoneg} as reference. 
 Hence, the open-shell ground state {\sBoneg} and the {\Btwog} state appear as a deexcitation and an excitation, respectively.
-With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below).
-
+With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below). 
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@@ -283,6 +279,8 @@ Hence, we rely on CC3/aug-cc-pVTZ to compute the equilibrium geometry of the {\o
 These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig.~\ref{fig:CBD}).
 The cartesian coordinates of all these geometries are provided in the {\SupInf}.
 Table \ref{tab:geometries} reports the key geometrical parameters obtained at these levels of theory as well as previous geometries computed by Manohar and Krylov at the CCSD(T)/cc-pVTZ level.
+%================================================
+
 
 %%% TABLE I %%%
 \begin{squeezetable}
@@ -312,9 +310,6 @@ Table \ref{tab:geometries} reports the key geometrical parameters obtained at th
 \end{squeezetable}
 %%% %%% %%% %%%
 
-%================================================
-
-
 %================================================
 \subsection{Autoisomerization barrier}
 \label{sec:auto}
@@ -351,7 +346,7 @@ CC3 & $6.59$ & $6.89$ & $7.88$ & $8.06$ \\
 CCSDT & $7.26$ & $7.64$ & $8.68$ &$\left[ 8.86\right]$\fnm[1]  \\
 CC4 & $7.40$ & $7.78$ & $\left[ 8.82\right]$\fnm[2] &  $\left[ 9.00\right]$\fnm[3]\\
 CCSDTQ & $7.51$ & $\left[ 7.89\right]$\fnm[4]& $\left[ 8.93\right]$\fnm[5]& $\left[ 9.11\right]$\fnm[6]\\
-\alert{CIPSI} & $7.91\pm 0.21$ & $8.58\pm 0.14$ &  &  \\
+%\alert{CIPSI} & $7.91\pm 0.21$ & $8.58\pm 0.14$ &  &  \\
 		\end{tabular}
 	\end{ruledtabular}
 	\fnt[1]{Value obtained using CCSDT/aug-cc-pVTZ corrected by the difference between CC3/aug-cc-pVQZ and CC3/aug-cc-pVTZ.}
@@ -378,34 +373,31 @@ The results concerning the autoisomerization barrier are reported in Table \ref{
 First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as \SI{10}{\kcalmol} for a given basis set between the different functionals.  
 Nonetheless, it is clear that the performance of a given functional is directly linked to the amount of exact exchange at short range.
 Indeed, hybrid functionals with a large fraction of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
-However, they are still off by \SIrange{1}{3}{\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 autoisomerization 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} 
 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{2.0}{\kcalmol} between different versions.
-In particular, we observe that SF-ADC(2)-s and SF-ADC(3) under- and overestimate the autoisomerization barrier, respectively, 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 autoisomerization 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).
 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.
 
 \alert{For the smaller active space, we have...}
 Concerning the multi-reference approaches with the (12e,12o) active space, we see a difference of the order of \SI{3}{\kcalmol} through all the bases between CASSCF and the second-order variants (CASPT2 and NEVPT2). 
 These differences can be explained by the well known lack of dynamical correlation at the CASSCF level. 
-The deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all the bases, CASPT2 being slightly more accurate than NEVPT2.
+The deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all bases, CASPT2 being slightly more accurate than NEVPT2.
 For each basis set, the CASPT2(12,12) and NEVPT2(12,12) are less than a \si{\kcalmol} away from the TBEs.
 
-\alert{Finally the last results shown in Table \ref{tab:auto_standard} are the CC ones, for the autoisomerization barrier energy we consider the CCSD, CCSDT, CCSDTQ methods and the approximations of CCSDT and of CCSDTQ, the CC3 and the CC4 methods, respectively. 
-We can see that the CCSD values are higher than the other CC methods with an energy difference of around \SIrange{1.05}{1.24}{\kcalmol} between the CCSD and the CCSDT methods. 
-The CCSDT and CCSDTQ autoisomerization barrier energies are closer with \SI{0.2}{\kcalmol} of energy difference. 
-The energy difference between the CCSDT and its approximation CC3 is about \SIrange{0.7}{0.8}{\kcalmol} for all the bases whereas the energy difference between the CCSDTQ and its approximate version CC4 is \SI{0.1}{\kcalmol}.}
-
+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 with the increase in computational cost of $\order*{N^6}$, $\order*{N^7}$, $\order*{N^8}$, $\order*{N^9}$, 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.
 
 %================================================
 
 %================================================
 \subsection{Excited States}
 \label{sec:states}
-%All the calculations are performed using four basis set, the 6-31+G(d) basis and the aug-cc-pVXZ with X$=$D, T, Q. In the following we will use the notation AVXZ for the aug-cc-pVXZ basis, again with X$=$D, T, Q. 
 
 \subsubsection{{\Dtwo} geometry}
 Table \ref{tab:sf_tddft_D2h} shows the results obtained for the vertical transition energies using spin-flip methods and Table \ref{tab:D2h} shows the results obtained with the standard methods. We discuss first 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}. 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. 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 bigger 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. 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. 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.