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almost done with CBD

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Pierre-Francois Loos 2021-01-18 17:09:57 +01:00
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Manuscript/sfBSE.tex
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@ -655,8 +655,8 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
% & (0.000) & 2.718(2.000) & 5.277(0.000) & 6.450(2.000) & 7.114(0.000) \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Excitation energy taken from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{Excitation energy taken from Ref.~\onlinecite{Krylov_2001a}.}
\fnt[1]{Excitation energies taken from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
\end{table*}
%\end{squeezetable}
%%% %%% %%% %%%
@ -726,31 +726,32 @@ Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\
\label{sec:CBD}
%===============================
Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Levchenko_2004,Manohar_2008,Karadakov_2008,Li_2009,Shen_2012,Lefrancois_2015,Casanova_2020,Vitale_2020}
Cyclobutadiene (CBD) is an interesting example as the electronic character of its ground state can be tuned via geometrical deformation. \cite{Balkova_1994,Levchenko_2004,Manohar_2008,Karadakov_2008,Li_2009,Shen_2012,Lefrancois_2015,Casanova_2020,Vitale_2020}
%with potential large spin contamination.
In its $D_{2h}$ rectangular $^1 A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals.
However, in its $D_{4h}$ square-planar $^3 A_{2g}$ ground-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals (\ie, singlet open-shell state).
In its $D_{2h}$ rectangular $A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals.
However, in its $D_{4h}$ square-planar $A_{2g}$ triplet state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals (\ie, singlet open-shell state).
In this case, single-reference methods notoriously fail.
Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations.
The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ states of CBD have been extracted from Ref.~\onlinecite{Manohar_2008} and have been obtained at the CCSD(T)/cc-pVTZ level.
EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}.
All of them have been obtained with a UHF reference like the SF-BSE calculations performed here.
Tables~\ref{tab:CBD_D2h} and \ref{tab:CBD_D4h} report excitation energies (with respect to the singlet ground state) obtained for the $D_{2h}$ and $D_{4h}$ geometries, respectively.
Tables~\ref{tab:CBD_D2h} and \ref{tab:CBD_D4h} report excitation energies (with respect to the singlet ground state) obtained at the $D_{2h}$ and $D_{4h}$ geometries, respectively, for several methods using the spin-flip \textit{ansatz}.
These are also represented in Fig.~\ref{fig:CBD}.
For each geometry, three states are under investigation.
For the $D_{2h}$ CBD, we look at the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$ and $2\,{}^1A_{1g}$ states.
In this case, it is important to mention that the $2\,{}^1A_{1g}$ state has a significant double excitation character.
For the $D_{4h}$ CBD we look at the $1\,{}^3 A_{2g}$, $2\,{}^1 A_{1g}$ and $1\,{}^1 B_{2g}$ states.
Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections.
In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, \ie, SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state.
This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state.
Adding dynamical corrections in SF-dBSE@{\GOWO} do not improve the accuracy of the excitation energies comparing to SF-ADC(3).
Indeed, we have a difference of 0.052 eV for the $1 ~^3 B_{1g}$ state, 0.393 eV for the $1 ~^1 B_{1g}$ state and 0.293 eV for the $2 ~^1 A_{1g}$ state.
Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-ADC(3) we have an interesting result.
Indeed, we have a wrong ordering in our first excited state, we find that the triplet state $1 ~^3 A_{2g}$ is lower in energy that the singlet state $B_{1g}$ in contrary to all of the results extracted in Refs.~\onlinecite{Manohar_2008,Lefrancois_2015}.
Then adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the difference of excitation energies with SF-ADC(3) it gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1\,{}^3 A_{2g}$ above the singlet state $B_{1g}$.
So here we have an example where the dynamical corrections are necessary to get the right chemistry.
For each geometry, three excited states are under investigation:
i) the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states of the $D_{2h}$ geometry;
ii) the $1\,{}^3 A_{2g}$, $2\,{}^1 A_{1g}$ and $1\,{}^1 B_{2g}$ states of the $D_{4h}$ geometry.
It is important to mention that the $2\,{}^1A_{1g}$ state of the rectangular geometry has a significant double excitation character, \cite{Loos_2019} and is then hardly described by second-order methods [such as CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} or EOM-CCSD \cite{Koch_1990,Stanton_1993,Koch_1994}] and remains a real challenge for third-order methods [as, for example, ADC(3), \cite{Trofimov_2002,Harbach_2014,Dreuw_2015} CC3, \cite{Christiansen_1995b} or EOM-CCSDT \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}].
Comparing the present SF-BSE@{\GOWO} results for the rectangular geometry (see Table \ref{tab:CBD_D2h}) to the most accurate ADC level, \ie, SF-ADC(3), we have a difference in the excitation energy of $0.017$ eV for the $1\,^3B_{1g}$ state.
This difference grows to $0.572$ eV for the $1\,^1B_{1g}$ state and then shrinks to $0.212$ eV for the $2\,^1A_{g}$ state.
Overall, adding dynamical corrections via the SF-dBSE@{\GOWO} scheme does not improve the accuracy of the excitation energies [as compared to SF-ADC(3)] with errors of $0.052$, $0.393$, and $0.293$ eV for the $1\,^3B_{1g}$, $1\,^1B_{1g}$, and $2\,^1 A_{g}$ states, respectively.
Now, looking at Table \ref{tab:CBD_D4h} which gathers the results for the square-planar geometry, we see that, at the SF-BSE@{\GOWO} level, the two first states are wrongly ordered with the triplet $1\,^3B_{1g}$ state lower than the singlet $1\,^1A_g$ state.
(The same observation can be made at the SF-TD-B3LYP level.)
This is certainly due to the poor Hartree-Fock reference and it could be potentially alleviated by using a better starting point of the $GW$ calculation.
Nonetheless, it is pleasing to see that adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the agreement in excitation energies with SF-ADC(3) but also gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1\,^3A_{2g}$ above the singlet state $B_{1g}$.
So here we have an example where the dynamical corrections are necessary to get the right state ordering.
%%% FIG 3 %%%
\begin{figure*}
@ -768,14 +769,14 @@ So here we have an example where the dynamical corrections are necessary to get
%%% TABLE ?? %%%
\begin{table}
\caption{
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
\label{tab:CBD_D2h}}
\begin{ruledtabular}
\begin{tabular}{lrrr}
& \mc{3}{c}{Excitation energies (eV)} \\
\cline{2-4}
Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\
Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
\hline
SF-TD-B3LYP\fnm[3] & $1.750$ & $2.260$ & $4.094$ \\
SF-TD-BH\&HLYP\fnm[3] & $1.583$ & $2.813$ & $4.528$ \\
@ -790,8 +791,8 @@ So here we have an example where the dynamical corrections are necessary to get
SF-dBSE@{\GOWO}\fnm[3] & $1.403$ & $2.883$ & $4.621$ \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015}.}
\fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.}
\fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.}
\fnt[3]{This work.}
\end{table}
%%% %%% %%% %%%
@ -821,8 +822,8 @@ So here we have an example where the dynamical corrections are necessary to get
SF-dBSE@{\GOWO}\fnm[3] & $0.012$ & $1.507$ & $1.841$ \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015}.}
\fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.}
\fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.}
\fnt[3]{This work.}
\end{table}
%%% %%% %%% %%%