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@ -577,12 +577,10 @@ In the following, all linear response calculations are performed within the TDA
%\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.} %\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
All the static and dynamic BSE calculations (labeled in the following as SF-BSE and SF-dBSE respectively) are performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}. All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(2)-x, respectively] as well as the SF-ADC(3) \cite{Lefrancois_2015} are performed with Q-CHEM 5.2.1. \cite{qchem4} The SF-ADC, EOM-SF-CC and SF-TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09}
Spin-flip TD-DFT calculations \cite{Shao_2003} considering the BLYP, \cite{Becke_1988,Lee_1988} B3LYP, \cite{Becke_1988,Lee_1988,Becke_1993a} and BH\&HLYP \cite{Lee_1988,Becke_1993b} functionals with contains $0\%$, $20\%$, and $50\%$ of exact exchange are labeled as SF-TD-BLYP, SF-TD-B3LYP, and SF-TD-BH\&HLYP, respectively, and are also performed with Q-CHEM 5.2.1. As a consistency check, we systematically perform the SF-CIS calculations with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09} Throughout this work, all spin-flip calculations have been performed with a UHF reference.
As a consistency check, we systematically perform the SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
Throughout this work, all spin-flip calculations are performed with a UHF reference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \section{Results}
@ -627,9 +625,9 @@ The excitation energies corresponding to the first singlet and triplet single ex
FCI\fnm[3] & 2.862 & 6.577 & 7.669 & 8.624 \\ FCI\fnm[3] & 2.862 & 6.577 & 7.669 & 8.624 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Values from Ref.~\onlinecite{Casanova_2020}.} \fnt[1]{Excitation energies extracted from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{This work.} \fnt[2]{This work.}
\fnt[3]{Values from Ref.~\onlinecite{Krylov_2001a}.} \fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
\end{table} \end{table}
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -639,7 +637,7 @@ The excitation energies corresponding to the first singlet and triplet single ex
\includegraphics[width=\linewidth]{Be} \includegraphics[width=\linewidth]{Be}
\caption{ \caption{
Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory: Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory:
SF-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black). SD-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black).
All the spin-flip calculations have been performed with a UHF reference. All the spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}} \label{fig:Be}}
\end{figure} \end{figure}
@ -754,31 +752,21 @@ The excitation energies corresponding to the first singlet and triplet single ex
\label{sec:H2} \label{sec:H2}
%=============================== %===============================
The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BHHLYP and BSE and are compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule. The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BHHLYP and BSE and are compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule.
Fig ~\ref{fig:H2_CIS} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing and the F${}^1 \Sigma_g^+$ state after the avoided crossing. SF-CIS does not give a good description of the double excitation. Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing and the F${}^1 \Sigma_g^+$ state after the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing and the F${}^1 \Sigma_g^+$ state is well describe before the avoided crossing. SF-CIS does not give a good description of the double excitation. The rigth panel gives results of the TD-BHHLYP calculation with and without spin-flip. TD-BHHLYP shows bad results for all the states of interest. In the last panel we have results for BSE calculation with and without spin-flip. SF-BSE gives a good representation of the B${}^1 \Sigma_u^+$ state with error of 0.05-0.3 eV. However SF-BSE does not describe well the E${}^1 \Sigma_g^+$ state with error of 0.5-1.6 eV.
\begin{figure}
\includegraphics[width=\linewidth]{H2_CIS.png}
\caption{
Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with SF-CIS and the dotted curves are obtained with CIS without using spin-flip.
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:H2_CIS}}
\end{figure}
\begin{figure} \begin{figure*}
\includegraphics[width=\linewidth]{H2_TDDFT.png} \includegraphics[width=0.49\linewidth]{H2_CIS.png}
\includegraphics[width=0.49\linewidth]{H2_TDDFT.png}
\includegraphics[width=0.49\linewidth]{H2_BSE_RHF.png}
\caption{ \caption{
Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with SF-TD-BHHLYP and the dotted curves are obtained with TD-BHHLYP without using spin-flip. Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with spin-flip method and the dotted curves are obtained without using spin-flip. The left panel shows CIS results, the right panel shows TD-BHHLYP results and the last panel shows the BSE results.
All the spin-flip calculations have been performed with a UHF reference. All the spin-flip calculations have been performed with a UHF reference.
\label{fig:H2_TDDFT}} \label{fig:H2}}
\end{figure} \end{figure*}
\begin{figure}
\includegraphics[width=\linewidth]{H2_BSE_RHF.png}
\caption{
Excitation energies of the three states of interest [with respect to the singlet ground state] of \ce{H2} obtained with the cc-pVQZ basis. Three sets of curves are drawn, the solid curves are the references (EOM-CCSD), the dashed curves are obtained with SF-BSE and the dotted curves are obtained with BSE with a RHF reference without using spin-flip.
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:H2_BSE}}
\end{figure}
%=============================== %===============================
\subsection{Cyclobutadiene} \subsection{Cyclobutadiene}
@ -799,8 +787,7 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
%%% TABLE ?? %%% %%% TABLE ?? %%%
\begin{table} \begin{table}
\caption{ \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 at the $D_{2h}$ rectangular equilibrium geometry of the $X\,{}^1 A_{g}$ singlet ground state. Vertical excitation energies (with respect to the singlet $X\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $X\,{}^1 A_{g}$ singlet ground state.
All the spin-flip calculations have been performed with a UHF reference.
\label{tab:CBD_D2h}} \label{tab:CBD_D2h}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccc} \begin{tabular}{lccc}
@ -815,12 +802,12 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
SF-ADC(2)-s\fnm[2] & 1.572& 3.201& 4.241\\ SF-ADC(2)-s\fnm[2] & 1.572& 3.201& 4.241\\
SF-ADC(2)-x\fnm[2] &1.576 &3.134 &3.792 \\ SF-ADC(2)-x\fnm[2] &1.576 &3.134 &3.792 \\
SF-ADC(3)\fnm[2] & 1.455&3.276 &4.328 \\ SF-ADC(3)\fnm[2] & 1.455&3.276 &4.328 \\
SF-BSE@{\GOWO}\fnm[3] & 1.438 & 2.704 &4.540 \\ SF-BSE@{\GOWO}@UHF\fnm[3] & 1.438 & 2.704 &4.540 \\
SF-dBSE@{\GOWO}\fnm[3] & 1.403 &2.883 &4.621 \\ SF-dBSE@{\GOWO}@UHF\fnm[3] & 1.403 &2.883 &4.621 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} \fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015}.} \fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.}
\fnt[3]{This work.} \fnt[3]{This work.}
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -828,8 +815,7 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
%%% TABLE ?? %%% %%% TABLE ?? %%%
\begin{table} \begin{table}
\caption{ \caption{
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $X\,{}^1B_{1g}$ singlet ground state. Vertical excitation energies (with respect to the singlet $X\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $X\,{}^1B_{1g}$ singlet ground state.
All the spin-flip calculations have been performed with a UHF reference.
\label{tab:CBD_D2h}} \label{tab:CBD_D2h}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccc} \begin{tabular}{lccc}
@ -844,12 +830,12 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
SF-ADC(2)-s\fnm[2] & & & \\ SF-ADC(2)-s\fnm[2] & & & \\
SF-ADC(2)-x\fnm[2] & & & \\ SF-ADC(2)-x\fnm[2] & & & \\
SF-ADC(3)\fnm[2] & & & \\ SF-ADC(3)\fnm[2] & & & \\
SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\ SF-BSE@{\GOWO}@UHF\fnm[3] & -0.049 & 1.189 & 1.480 \\
SF-dBSE@{\GOWO}\fnm[3] & 0.012 & 1.507 & 1.841 \\ SF-dBSE@{\GOWO}@UHF\fnm[3] & 0.012 & 1.507 & 1.841 \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} \fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015}.} \fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.}
\fnt[3]{This work.} \fnt[3]{This work.}
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%