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Manuscript/sfBSE.bib
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@ -1,13 +1,37 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-10 22:26:03 +0100
%% Created for Pierre-Francois Loos at 2021-01-11 09:47:39 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Koch_1994,
author = {Koch,Henrik and Kobayashi,Rika and Sanchez de Mer{\'a}s,Alfredo and Jorgensen, Poul},
date-added = {2021-01-11 09:32:50 +0100},
date-modified = {2021-01-11 09:32:50 +0100},
doi = {10.1063/1.466321},
journal = {J. Chem. Phys.},
pages = {4393-4400},
title = {Calculation of size‐intensive transition moments from the coupled cluster singles and doubles linear response function},
volume = {100},
year = {1994},
Bdsk-Url-1 = {https://doi.org/10.1063/1.466321}}
@article{Stanton_1993,
author = {Stanton,John F. and Bartlett,Rodney J.},
date-added = {2021-01-11 09:32:39 +0100},
date-modified = {2021-01-11 09:32:39 +0100},
doi = {10.1063/1.464746},
journal = {J. Chem. Phys.},
pages = {7029-7039},
title = {The equation of motion coupled‐cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties},
volume = {98},
year = {1993},
Bdsk-Url-1 = {https://doi.org/10.1063/1.464746}}
@article{Sarkar_2021,
author = {R. Sarkar and M. Boggio-Pasqua and P. F. Loos and D. Jacquemin},
date-added = {2021-01-10 18:27:30 +0100},
@ -245,10 +269,10 @@
year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00289}}
@article{Chrayteh_2020,
@article{Chrayteh_2021,
author = {Chrayteh, Amara and Blondel, Aymeric and Loos, Pierre-Fran{\c c}ois and Jacquemin, Denis},
date-added = {2020-12-09 09:59:26 +0100},
date-modified = {2020-12-09 09:59:26 +0100},
date-modified = {2021-01-11 09:19:56 +0100},
doi = {10.1021/acs.jctc.0c01111},
journal = {Journal of Chemical Theory and Computation},
number = {0},
@ -3487,44 +3511,25 @@
@article{Holzer_2018a,
author = {Holzer,Christof and Klopper,Wim},
date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200},
date-modified = {2021-01-11 09:18:51 +0100},
doi = {10.1063/1.5051028},
eprint = {https://doi.org/10.1063/1.5051028},
journal = {J. Chem. Phys.},
number = {10},
pages = {101101},
title = {Communication: A hybrid Bethe--Salpeter/time-dependent density-functional-theory approach for excitation energies},
url = {https://doi.org/10.1063/1.5051028},
volume = {149},
year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5051028}}
@article{Holzer_2018b,
author = {Holzer, Christof and Gui, Xin and Harding, Michael E. and Kresse, Georg and Helgaker, Trygve and Klopper, Wim},
date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200},
doi = {10.1063/1.5047030},
eprint = {https://doi.org/10.1063/1.5047030},
journal = {J. Chem. Phys.},
number = {14},
pages = {144106},
title = {Bethe-Salpeter correlation energies of atoms and molecules},
url = {https://doi.org/10.1063/1.5047030},
volume = {149},
year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5047030}}
@article{Holzer_2019,
author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim},
date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200},
date-modified = {2021-01-11 09:18:12 +0100},
doi = {10.1063/1.5093396},
eprint = {https://doi.org/10.1063/1.5093396},
journal = {J. Chem. Phys.},
number = {21},
pages = {214112},
title = {GW Quasiparticle Energies of Atoms in Strong Magnetic Fields},
url = {https://doi.org/10.1063/1.5093396},
volume = {150},
year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5093396}}
@ -4538,15 +4543,12 @@
@article{Stein_2009,
author = {Stein, Tamar and Kronik, Leeor and Baer, Roi},
date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200},
date-modified = {2021-01-11 09:20:50 +0100},
doi = {10.1021/ja8087482},
eprint = {https://doi.org/10.1021/ja8087482},
journal = {J. Am. Chem. Soc.},
note = {PMID: 19239266},
number = {8},
pages = {2818-2820},
title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory},
url = {https://doi.org/10.1021/ja8087482},
volume = {131},
year = {2009},
Bdsk-Url-1 = {https://doi.org/10.1021/ja8087482}}
@ -4996,10 +4998,10 @@
year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1002/cphc.201100200}}
@article{Holzer_2018,
@article{Holzer_2018b,
author = {Christof Holzer and Xin Gui and Michael E. Harding and Georg Kresse and Trygve Helgaker and Wim Klopper},
date-added = {2020-01-04 20:49:55 +0100},
date-modified = {2020-02-05 20:58:26 +0100},
date-modified = {2021-01-11 09:18:02 +0100},
doi = {10.1063/1.5047030},
journal = {J. Chem. Phys.},
pages = {144106},
@ -8699,12 +8701,9 @@
@article{Koch_1990,
author = {Koch, Henrik and Jensen, Hans Jo/rgen Aa. and Jo/rgensen, Poul and Helgaker, Trygve},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-01-01 21:36:52 +0100},
date-modified = {2021-01-11 09:32:58 +0100},
doi = {10.1063/1.458815},
issn = {0021-9606, 1089-7690},
journal = {J. Chem. Phys.},
language = {en},
month = sep,
number = {5},
pages = {3345-3350},
title = {Excitation Energies from the Coupled Cluster Singles and Doubles Linear Response Function ({{CCSDLR}}). {{Applications}} to {{Be}}, {{CH}} {\textsuperscript{+}} , {{CO}}, and {{H}} {\textsubscript{2}} {{O}}},
@ -9006,19 +9005,6 @@
volume = {478},
year = {2009}}
@article{Levine_2006a,
author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Mart\'Inez, Todd J.},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-01-01 21:36:52 +0100},
doi = {10.1080/00268970500417762},
journal = {Mol. Phys.},
month = mar,
pages = {1039-1051},
title = {Conical Intersections and Double Excitations in Time-Dependent Density Functional Theory},
volume = {104},
year = {2006},
Bdsk-Url-1 = {https://doi.org/10.1080/00268970500417762}}
@article{Levy_1995,
author = {Levy, Mel},
date-added = {2020-01-01 21:36:51 +0100},
@ -12828,14 +12814,11 @@
Bdsk-Url-1 = {https://doi.org/10.1063/1.1633756}}
@article{Levine_2006,
author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Mart\'Inez, Todd J.},
author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Martinez, Todd J.},
date-added = {2020-01-01 21:36:32 +0100},
date-modified = {2020-01-01 21:36:32 +0100},
date-modified = {2021-01-11 09:19:29 +0100},
doi = {10.1080/00268970500417762},
issn = {0026-8976, 1362-3028},
journal = {Mol. Phys.},
language = {en},
month = mar,
number = {5-7},
pages = {1039-1051},
title = {Conical Intersections and Double Excitations in Time-Dependent Density Functional Theory},

52
Manuscript/sfBSE.tex
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@ -577,10 +577,12 @@ 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 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.
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
@ -606,7 +608,7 @@ 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 & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$
Method & $^3P(1s^22s^12p^1)$ & $^1P(1s^22s^12p^1)$
& $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\
\hline
SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\
@ -625,9 +627,9 @@ 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]{Excitation energies extracted from Ref.~\onlinecite{Casanova_2020}.}
\fnt[1]{Values from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{This work.}
\fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
\fnt[3]{Values from Ref.~\onlinecite{Krylov_2001a}.}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
@ -637,7 +639,7 @@ 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:
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).
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).
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}}
\end{figure}
@ -751,7 +753,15 @@ 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}
@ -795,7 +805,8 @@ 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 $X\,{}^1A_{g}$ 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 $\text{X}\,{}^1A_{g}$ 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.
\label{tab:CBD_D2h}}
\begin{ruledtabular}
\begin{tabular}{lccc}
@ -810,12 +821,17 @@ 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] & 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]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.}
\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}
%%% %%% %%% %%%
@ -823,8 +839,9 @@ 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 $X\,{}^1B_{1g}$ singlet ground state.
\label{tab:CBD_D2h}}
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}}
\begin{ruledtabular}
\begin{tabular}{lccc}
& \mc{3}{c}{Excitation energies (eV)} \\
@ -838,12 +855,17 @@ 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]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.}
\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}
%%% %%% %%% %%%

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