diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index e437489..be02867 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -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.} 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}. -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} -Spin-flip TD-DFT calculations \cite{Shao_2003} considering the BLYP, B3LYP, and BH\&HLYP 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. -EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09} -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. +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 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} +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. +Throughout this work, all spin-flip calculations have been performed with a UHF reference. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} @@ -608,7 +606,7 @@ The excitation energies corresponding to the first singlet and triplet single ex \begin{tabular}{lcccc} & \mc{4}{c}{Excitation energies (eV)} \\ \cline{2-5} - Method & $^3P(1s^22s^12p^1)$ & $^1P(1s^22s^12p^1)$ + Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$ & $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\ \hline SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\ @@ -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 \\ \end{tabular} \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[3]{Values from Ref.~\onlinecite{Krylov_2001a}.} + \fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.} \end{table} \end{squeezetable} %%% %%% %%% %%% @@ -639,7 +637,7 @@ The excitation energies corresponding to the first singlet and triplet single ex \includegraphics[width=\linewidth]{Be} \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: - 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. \label{fig:Be}} \end{figure} @@ -753,28 +751,13 @@ The excitation energies corresponding to the first singlet and triplet single ex \subsection{Hydrogen molecule} \label{sec:H2} %=============================== -<<<<<<< HEAD -The second system of interest is the \ce{H2} molecule where we streched the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states were investigated during the streching: 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. -======= -Our second example deals with the dissociation of the \ce{H2} molecule. -The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $1\sigma_g^2$, and the lowest singlet excited state is of $\text{B}\,{}^1 \Sigma_u^+$ symmetry with a configuration $1\sigma_g 1\sigma_u$. -Of particular interest here are the two lowest singlet excited states of $\Sigma_g^+$ spatial symmetry: the singly excited $\text{E}\,{}^1 \Sigma_g^+$ with configuration $1\sigma_g 2\sigma_g$, and the doubly excited $\text{F}\,{}^1 \Sigma_g^+$ of configuration $1\sigma_u^2$ . -Because these two excited states interact strongly and form an avoided crossing around $R_{\ce{H-H}} = 1.4$ \AA, they are usually labeled as the $\text{EF}\,{}^1 \Sigma_g^+$ state. - ->>>>>>> ec35e13a169105bfc5b5f7b6435e4aab238b63e6 - -\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 reference (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} +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. \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 reference (EOM-CCSD), the dashed curves are obtained with SF-CIS and the dotted curves are obtained with CIS 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 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} @@ -782,10 +765,19 @@ Because these two excited states interact strongly and form an avoided crossing \begin{figure} \includegraphics[width=\linewidth]{H2_TDDFT.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 reference (EOM-CCSD), the dashed curves are obtained with SF-TDBHHLYP and the dotted curves are obtained with TDBHHLYP 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 SF-TD-BHHLYP and the dotted curves are obtained with TD-BHHLYP without using spin-flip. All the spin-flip calculations have been performed with a UHF reference. \label{fig:H2_TDDFT}} \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} \label{sec:CBD} @@ -805,8 +797,7 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations %%% 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 at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ singlet ground state. - All the spin-flip calculations have been performed with a UHF reference. + 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. \label{tab:CBD_D2h}} \begin{ruledtabular} \begin{tabular}{lccc} @@ -818,20 +809,15 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations EOM-SF-CCSD\fnm[1] & & & \\ EOM-SF-CCSD(fT)\fnm[1] & & & \\ EOM-SF-CCSD(dT)\fnm[1] & & & \\ - SF-ADC(2)-s\fnm[2] & & & \\ - SF-ADC(2)-x\fnm[2] & & & \\ - SF-ADC(3)\fnm[2] & & & \\ -<<<<<<< HEAD + 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(3)\fnm[2] & 1.455&3.276 &4.328 \\ SF-BSE@{\GOWO}@UHF\fnm[3] & 1.438 & 2.704 &4.540 \\ SF-dBSE@{\GOWO}@UHF\fnm[3] & 1.403 &2.883 &4.621 \\ -======= - SF-BSE@{\GOWO}\fnm[3] & & & \\ - SF-dBSE@{\GOWO}\fnm[3] & & & \\ ->>>>>>> ec35e13a169105bfc5b5f7b6435e4aab238b63e6 \end{tabular} \end{ruledtabular} - \fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} - \fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.} + \fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.} + \fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.} \fnt[3]{This work.} \end{table} %%% %%% %%% %%% @@ -839,9 +825,8 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations %%% TABLE ?? %%% \begin{table} \caption{ - 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 $\text{X}\,{}^1B_{1g}$ singlet ground state. - All the spin-flip calculations have been performed with a UHF reference. - \label{tab:CBD_D4h}} + 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. + \label{tab:CBD_D2h}} \begin{ruledtabular} \begin{tabular}{lccc} & \mc{3}{c}{Excitation energies (eV)} \\ @@ -855,17 +840,12 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations SF-ADC(2)-s\fnm[2] & & & \\ SF-ADC(2)-x\fnm[2] & & & \\ SF-ADC(3)\fnm[2] & & & \\ -<<<<<<< HEAD SF-BSE@{\GOWO}@UHF\fnm[3] & -0.049 & 1.189 & 1.480 \\ SF-dBSE@{\GOWO}@UHF\fnm[3] & 0.012 & 1.507 & 1.841 \\ -======= - SF-BSE@{\GOWO}\fnm[3] & & & \\ - SF-dBSE@{\GOWO}\fnm[3] & & & \\ ->>>>>>> ec35e13a169105bfc5b5f7b6435e4aab238b63e6 \end{tabular} \end{ruledtabular} - \fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} - \fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.} + \fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.} + \fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.} \fnt[3]{This work.} \end{table} %%% %%% %%% %%%