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Manuscript/CBD.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-06-08 17:31:37 +0200
%% Created for Pierre-Francois Loos at 2022-06-08 18:49:54 +0200
%% Saved with string encoding Unicode (UTF-8)

29
Manuscript/CBD.tex
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@ -203,7 +203,7 @@ For ionic excited states, like the {\sBoneg} state of CBD, it is particularly im
On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
In order to avoid the intruder state problem in CASPT2, a real-valued level shift of \SI{0.3}{\hartree} is set, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.25}{\hartree} to avoid systematic underestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022}
\alert{For the sake of comparison, in some cases, we have also performed multi-reference CI calculations including Davidson correction (MRCI+Q). \cite{Knowles_1988,Werner_1988}}
\alert{For the sake of comparison, for the (4e,4o) active space, we have also performed multi-reference CI calculations including Davidson correction (MRCI+Q). \cite{Knowles_1988,Werner_1988}}
All these calculations are carried out with MOLPRO. \cite{Werner_2020}
%and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -227,7 +227,7 @@ Additionally, we have also computed SF-TD-DFT excitation energies using range-se
Finally, the hybrid meta-GGA functional M06-2X (54\% of exact exchange) \cite{Zhao_2008} and the RSH meta-GGA functional M11 (42.8\% of short-range exact exchange and 100\% at long range) \cite{Peverati_2011} are also employed.
Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
\alert{There also exist spin-flip extensions of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} and we consider here the spin-flip version of EOM-CCSD, named EOM-SF-CCSD. \cite{Krylov_2001a}
\alert{There also exist spin-flip extensions of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} and we consider here the spin-flip version of EOM-CCSD, named SF-EOM-CCSD. \cite{Krylov_2001a}
Additionally, Manohar and Krylov introduced a non-iterative triples correction to EOM-CCSD and extended it to the spin-flip variant. \cite{Manohar_2008}
Two types of triples corrections were proposed: (i) EOM-CCSD(dT) that uses the diagonal elements of the similarity-transformed CCSD Hamiltonian, and (ii) EOM-CCSD(fT) where the Hartree-Fock orbital energies are considered instead.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -355,7 +355,7 @@ SF-ADC(2)-s & $6.69$ & $6.98$ & $8.63$ & \\
SF-ADC(2)-x & $8.63$ & $8.96$ &$10.37$ & \\
SF-ADC(2.5) & $7.36$ & $7.76$ & $9.11$ & \\
SF-ADC(3) & $8.03$ & $8.54$ & $9.58$ \\
\alert{EOM-SF-CCSD} & \alert{$5.86$} & \alert{$6.27$} & \alert{$7.40$} \\[0.1cm]
\alert{SF-EOM-CCSD} & \alert{$5.86$} & \alert{$6.27$} & \alert{$7.40$} \\[0.1cm]
CASSCF(4,4) & $6.17$ & $6.59$ & $7.38$ & $7.41$ \\
CASPT2(4,4) & $6.56$ & $6.87$ & $7.77$ & $7.93$ \\
SC-NEVPT2(4,4) & $7.95$ & $8.31$ & $9.23$ & $9.42$ \\
@ -409,7 +409,12 @@ We note that SF-ADC(2)-x [which scales as $\order*{N^6}$] is probably not worth
This behavior was previously reported by Dreuw's group. \cite{Wormit_2014,Harbach_2014,Dreuw_2015}
Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models.
\alert{We observe that EOM-SF-CCSD tends to underestimate of about \SI{1.5}{\kcalmol} the energy barrier compared to the TBE. This can be amended by using the triples correction with the EOM-SF-CCSD(fT) and EOM-SF-CCSD(dT) methods (see {\SupInf}). We also note that the EOM-SF-CCSD values for the energy barrier are close to the CC3 ones. Our results are in agreement with previous studies \cite{Manohar_2008,Lefrancois_2015} and justify to avoid the more expensive calculations of the triples correction as we can expect a similar trend. Note that contrary to a previous statement \cite{Manohar_2008} the (fT) correction performs better than the (dT) one for the energy barrier (however, for the excited states we retrieve the same statement).}
\alert{We observe that SF-EOM-CCSD tends to underestimate by about \SI{1.5}{\kcalmol} the energy barrier compared to the TBE.
This can be alleviated by including the triples correction with SF-EOM-CCSD(fT) and SF-EOM-CCSD(dT) (see {\SupInf}).
We also note that the SF-EOM-CCSD values for the energy barrier are close to the CC3 ones.
Our results are in agreement with previous studies \cite{Manohar_2008,Lefrancois_2015} and justify to avoid the more expensive calculations of the triples correction as we can expect a similar trend.
Note that contrary to a previous statement \cite{Manohar_2008} the (fT) correction performs better than the (dT) correction for the energy barrier.
However, for the excited states, we retrieve the same statement (see below).}
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.
@ -497,7 +502,7 @@ SF-ADC(2.5) & 6-31+G(d) & $1.496$ & $3.328$ & $4.219$ \\
SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
& aug-cc-pVDZ & $1.422$ & $3.180$ & $4.208$ \\
& aug-cc-pVTZ & $1.419$ & $3.162$ & $4.224$ \\[0.1cm]
\alert{EOM-SF-CCSD} & \alert{6-31+G(d)} & \alert{$1.663$} & \alert{$3.515$} & \alert{$4.275$} \\
\alert{SF-EOM-CCSD} & \alert{6-31+G(d)} & \alert{$1.663$} & \alert{$3.515$} & \alert{$4.275$} \\
& \alert{aug-cc-pVDZ} & \alert{$1.611$} & \alert{$3.315$} & \alert{$4.216$} \\
& \alert{aug-cc-pVTZ} & \alert{$1.609$} & \alert{$3.293$} & \alert{$4.245$} \\[0.1cm]
%SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
@ -635,9 +640,9 @@ This further motivates the ``pyramidal'' extrapolation scheme that we have emplo
Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the present context with much larger errors than the other schemes.
Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns making SF-ADC(2.5) particularly accurate except for the doubly-excited state {\twoAg} where the error with respect to the TBE (\SI{0.140}{\eV}) is larger than the SF-ADC(2)-s error (\SI{0.093}{\eV}).
\alert{We observe that EOM-SF-CCSD excitation energies are closer to the SF-ADC(2)-s, with an energy difference of about \SI{0.1}{\eV}, than the other schemes as it was already noticed by LeFrançois and co-workers. \cite{Lefrancois_2015}
We see that the EOM-SF-CCSD excitations energies for the triplet state are larger of about (\SI{0.3}{\eV} compared to the CCSD ones, this was also present in the study of Manohar and Krylov. \cite{Manohar_2008}
Again, we have similar results, with EOM-SF-CCSD, than previous studies \cite{Manohar_2008,Lefrancois_2015} for the excited states. We can logically expect similar trend for EOM-SF-CCSD(fT) and EOM-SF-CCSD(dT) that lower the excitation energies and tend to be in a better agreement with respect to the TBE (see {\SupInf}). Note that the (dT) correction demonstrates better performance than the (fT) one as previously observed. \cite{Manohar_2008}}
\alert{We observe that SF-EOM-CCSD excitation energies are closer to the SF-ADC(2)-s, with an energy difference of about \SI{0.1}{\eV}, than the other schemes as it was already noticed by LeFrançois and co-workers. \cite{Lefrancois_2015}
We see that the SF-EOM-CCSD excitations energies for the triplet state are larger of about (\SI{0.3}{\eV} compared to the CCSD ones, this was also present in the study of Manohar and Krylov. \cite{Manohar_2008}
Again, we have similar results, with SF-EOM-CCSD, than previous studies \cite{Manohar_2008,Lefrancois_2015} for the excited states. We can logically expect similar trend for SF-EOM-CCSD(fT) and SF-EOM-CCSD(dT) that lower the excitation energies and tend to be in a better agreement with respect to the TBE (see {\SupInf}). Note that the (dT) correction demonstrates better performance than the (fT) one as previously observed. \cite{Manohar_2008}}
Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table \ref{tab:D2h}.
Regarding the \alert{multi-reference} 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.
@ -717,7 +722,7 @@ SF-ADC(2.5) & 6-31+G(d) & $0.234$ & $1.705$ & $2.087$ \\
SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
& aug-cc-pVDZ & $0.088$ & $1.571$ & $1.878$ \\
& aug-cc-pVTZ & $0.079$ & $1.575$ & $1.853$ \\[0.1cm]
\alert{EOM-SF-CCSD} & \alert{6-31+G(d)} & \alert{$0.446$} & \alert{$1.875$} & \alert{$2.326$} \\
\alert{SF-EOM-CCSD} & \alert{6-31+G(d)} & \alert{$0.446$} & \alert{$1.875$} & \alert{$2.326$} \\
& \alert{aug-cc-pVDZ} & \alert{$0.375$} & \alert{$1.776$} & \alert{$2.102$} \\
& \alert{aug-cc-pVTZ}& \alert{$0.354$} & \alert{$1.768$} & \alert{$2.060$} \\
\end{tabular}
@ -845,7 +850,7 @@ Although it provides a decent singlet-triplet gap value, SF-ADC(2)-x seems to pa
Again, averaging the SF-ADC(2)-s and SF-ADC(3) transition energies is beneficial in most cases at the exception of {\Aoneg}.
Although the basis set effects are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and this holds for wave function methods in general.
\alert{Again, the EOM-SF-CCSD excitation energies are closer to the SF-ADC(2)-s ones than the other schemes and (dT) and (fT) corrections tend to give a better agreement with respect to the TBE (see {\SupInf}). As for the {\Dtwo} excitation energies, the (dT) correction performs better than the (fT) one.}
\alert{Again, the SF-EOM-CCSD excitation energies are closer to the SF-ADC(2)-s ones than the other schemes and (dT) and (fT) corrections tend to give a better agreement with respect to the TBE (see {\SupInf}). As for the {\Dtwo} excitation energies, the (dT) correction performs better than the (fT) one.}
Let us turn to the multi-reference results (Table \ref{tab:D4h}).
For both active spaces, expectedly, CASSCF does not provide a quantitive energetic description, although it is worth mentioning that the right state ordering is preserved.
@ -881,8 +886,8 @@ This has been shown to be clearly beneficial for the automerization barrier and
\item At the SF-ADC level, we have found that, as expected, 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) emerges as an excellent compromise.
\item \alert{EOM-SF-CCSD shows similar performance than cheaper SF-ADC(2)-s formalism, especially for the excitation energies.
As previously reported, the two variants including non-iterative triples corrections, EOM-SF-CCSD(dT) and EOM-SF-CCSD(fT), improve the results, the (dT) correction performing slightly better for the vertical excitation energies computed at the {\Dtwo} and {\Dfour} equilibrium geometries.}
\item \alert{SF-EOM-CCSD shows similar performance than the cheaper SF-ADC(2)-s formalism, especially for the excitation energies.
As previously reported, the two variants including non-iterative triples corrections, SF-EOM-CCSD(dT) and SF-EOM-CCSD(fT), improve the results, the (dT) correction performing slightly better for the vertical excitation energies computed at the {\Dtwo} and {\Dfour} equilibrium geometries.}
\item For the {\Dfour} square planar structure, a faithful energetic description of the excited states is harder to reach at the SF-TD-DFT level because of the strong multi-configurational character.
In such scenario, the SF-TD-DFT excitation energies can exhibit errors of the order of \SI{1}{\eV} compared to the TBEs.

18
Manuscript/sup-CBD.tex
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@ -163,7 +163,7 @@ H -2.092429 0.000000 0.000000
\section{Comment about symmetry: standard vs non-standard orientation}
%%%%%%%%%%%%%%%%%%%%%%%%
At the $D_{4h}$ $T_1$ optimized geometry, we have used the conventional standard orientation where two $C_2$ axes run through the carbon atoms.
\alert{At the $D_{4h}$ $T_1$ optimized geometry, we have used the conventional standard orientation where two $C_2$ axes run through the carbon atoms.
In this conventional orientation, the singlet ground state $1 ^1B_{1g}$ remains $1 ^1B_{1g}$ in the $D_{2h}$ point group and the singlet excited state $1 ^1A_{1g}$ becomes $1 ^1A_g$ in the $D_{2h}$ point group.
Upon rotation of the molecular framework by 45 degrees in the $xy$ plane, the two $C_2$ axes then bisect the carbon-carbon bonds.
This induces a different orbital picture.
@ -179,7 +179,7 @@ The $1 ^1B_{1g}$ ground state is obtained as a singly excited state from that re
In the other (non-standard) orientation, the lowest $^1A_g$ state correlates with the $1 ^1B_{1g}$ ground state, which in this orientation has a strong double-excitation character.
Then, the $1 ^1A_{1g}$ excited state has also a strong double-excitation character, while the $1 ^1B_{2g}$ excited state is obtained by one-electron excitation.
Thus, whatever the orientation of the molecule, we will face the same problem for the reference state.
Note that in the case of the SF formalism, these three singlet states should all be described correctly if one takes the $1 ^3A_{2g}$ state as a reference high spin state, whatever the orientation.
Note that in the case of the SF formalism, these three singlet states should all be described correctly if one takes the $1 ^3A_{2g}$ state as a reference high spin state, whatever the orientation.}
\begin{figure}
\includegraphics[width=\textwidth]{MOs}
@ -214,7 +214,7 @@ SF-ADC(2)-s & $-0.30$ & \alert{$0.098$} & $-0.026$ & $0.093$ & $0.112$ & $0.112$
SF-ADC(2)-x & $1.44$ & \alert{$0.106$} & $-0.094$ & $-0.335$ & $0.068$ & $-0.409$ & $-0.118$ \\
SF-ADC(2.5) & $0.18$ & \alert{$0.042$} & $0.006$ &$0.140$ & $0.024$ & $0.094$ & $0.000$ \\
SF-ADC(3) & $0.65$ & \alert{$-0.014$} & $0.037$ & $0.186$ & $-0.065$ & $0.075$ & $0.004$ \\
\alert{EOM-SF-CCSD} & \alert{$-1.53$} & \alert{$0.176$} & \alert{$0.168$} & \alert{$0.207$} & \alert{$0.210$} & \alert{$0.268$} & \alert{$0.211$} \\[0.1cm]
\alert{SF-EOM-CCSD} & \alert{$-1.53$} & \alert{$0.176$} & \alert{$0.168$} & \alert{$0.207$} & \alert{$0.210$} & \alert{$0.268$} & \alert{$0.211$} \\[0.1cm]
CASSCF(4,4) & $-1.55$ & \alert{$0.237$} & $1.421$ & $0.403$& $0.290$ & $0.734$ & $1.575$ \\
CASPT2(4,4) & $-1.16$ & \alert{$-0.021$} & $-0.202$ & $0.034$ & $-0.016$ & $0.006$ & $-0.214$ \\
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
@ -252,12 +252,12 @@ Literature & $8.53$\fnm[3] & $1.573$\fnm[3] & $3.208$\fnm[3] & $4.247$\fnm[3] &
\fnt[3]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-s/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[4]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-x/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[5]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(3)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[6]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[7]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD(fT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[8]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the EOM-SF-CCSD(dT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[9]{Value obtained from Ref.~\onlinecite{Gulania_2021} at the EOM-DEA-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[10]{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(3p-1h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}
\fnt[11]{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(4p-2h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}
\fnt[6]{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[7]{\alert{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD(fT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[8]{\alert{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD(dT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[9]{\alert{Value obtained from Ref.~\onlinecite{Gulania_2021} at the EOM-DEA-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[10]{\alert{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(3p-1h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}}
\fnt[11]{\alert{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(4p-2h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}}
\end{table*}
%\end{squeezetable}

4
Response_Letter/Response_Letter.tex
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@ -29,7 +29,9 @@ We look forward to hearing from you.
%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}
{This article presents a survey of spin-flip TD-DFT, spin-flip ADC, multireference (CASSCF and MRPT), and equation-of-motion coupled cluster methods as applied to the automerization and vertical excitation energies of cyclobutadiene (CBD). As the smallest example of anti-aromaticity (and one of the smallest and most interesting exemplars of strong PJT distortion), CBD is an illuminating and challenging test case for these methods. (EOM-)CCSDTQ values, with a “pyramidal” basis set extrapolation scheme are used as the newly-proposed theoretical best estimates, and limited selected full CI (CIPSI) calculations confirm their excellent accuracy. The authors reach some interesting and useful conclusions concerning the tested methods.
{This article presents a survey of spin-flip TD-DFT, spin-flip ADC, multireference (CASSCF and MRPT), and equation-of-motion coupled cluster methods as applied to the automerization and vertical excitation energies of cyclobutadiene (CBD).
As the smallest example of anti-aromaticity (and one of the smallest and most interesting exemplars of strong PJT distortion), CBD is an illuminating and challenging test case for these methods. (EOM-)CCSDTQ values, with a “pyramidal” basis set extrapolation scheme are used as the newly-proposed theoretical best estimates, and limited selected full CI (CIPSI) calculations confirm their excellent accuracy.
The authors reach some interesting and useful conclusions concerning the tested methods.
}
\\
\alert{