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Manuscript/sfBSE.tex
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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.}
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}
%%% %%% %%% %%%