diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 642ae94..317dbee 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -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} %%% %%% %%% %%%