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Manuscript/sfBSE.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-10 18:27:42 +0100
%% Created for Pierre-Francois Loos at 2021-01-10 22:26:03 +0100
%% Saved with string encoding Unicode (UTF-8)
@ -1492,23 +1492,6 @@
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}}
@article{Krylov_2001b,
abstract = {A new approach to the bond-breaking problem is proposed. Both closed and open shell singlet states are described within a single reference formalism as spin-flipping, e.g., α→β, excitations from a triplet (Ms=1) reference state for which both dynamical and non-dynamical correlation effects are much smaller than for the corresponding singlet state. Formally, the new theory can be viewed as an equation-of-motion (EOM) model where excited states are sought in the basis of determinants conserving the total number of electrons but changing the number of α and β electrons. The results for two simplest members of the proposed hierarchy of approximations are presented.},
author = {Anna I. Krylov},
date-added = {2020-12-06 14:36:46 +0100},
date-modified = {2020-12-06 14:37:01 +0100},
doi = {https://doi.org/10.1016/S0009-2614(01)00287-1},
issn = {0009-2614},
journal = {Chem. Phys. Lett.},
number = {4},
pages = {375 - 384},
title = {Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model},
url = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
volume = {338},
year = {2001},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871},
Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}}
@article{Krylov_2002,
author = {Krylov,Anna I. and Sherrill,C. David},
date-added = {2020-12-06 14:36:32 +0100},
@ -4204,9 +4187,9 @@
@article{Refaely-Abramson_2012,
author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik},
date-added = {2020-05-18 21:40:28 +0200},
date-modified = {2020-05-18 21:40:28 +0200},
date-modified = {2021-01-10 20:59:42 +0100},
doi = {10.1103/PhysRevLett.109.226405},
journal = {Phys. Rev. X},
journal = {Phys. Rev. Lett.},
pages = {226405},
title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional},
volume = {109},
@ -8832,10 +8815,10 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.075143},
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.93.075143}}
@article{Krylov_2001c,
@article{Krylov_2001b,
author = {Krylov, Anna I.},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-12-06 14:37:24 +0100},
date-modified = {2021-01-10 22:26:03 +0100},
doi = {10.1016/S0009-2614(01)01316-1},
issn = {00092614},
journal = {Chem. Phys. Lett.},

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Manuscript/sfBSE.tex
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@ -572,13 +572,15 @@ Note that, in any case, the entire set of orbitals and energies is corrected.
Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies.
This is left for future work.
However, it is worth mentioning that, for the present (small) molecular systems, HF is usually an excellent starting point. \cite{Loos_2020a,Loos_2020e,Loos_2020h}
However, it is worth mentioning that, for the present (small) molecular systems, HF is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012}
In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
\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.
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 TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09}
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}
@ -589,40 +591,56 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
\subsection{Beryllium atom}
\label{sec:Be}
%===============================
As a first example, we consider the simple case of the beryllium atom which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020}
Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration.
The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $P$ and $D$ spatial symmetries (respectively), obtained with the 6-31G basis set are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}.
%%% TABLE I %%%
\begin{squeezetable}
\begin{table}
\caption{
Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained for various methods with the 6-31G basis.
All the spin-flip calculations have been performed with a UHF reference.
\label{tab:Be}}
\begin{ruledtabular}
\begin{tabular}{lcccc}
& \mc{4}{c}{Excitation energies (eV)} \\
& \mc{4}{c}{Excitation energies (eV)} \\
\cline{2-5}
Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$
& $^3P(1s^22p^2)$ & $^1P(1s^22p^2)$ \\
Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$
& $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\
\hline
SF-TD-BLYP & 3.210 & 3.210 & 6.691 & 7.598 \\
SF-TD-B3LYP & 3.332 & 4.275 & 6.864 & 7.762 \\
SF-TD-BHHLYP & 2.874 & 4.922 & 7.112 & 8.188 \\
SF-CIS & 2.111 & 6.036 & 7.480 & 8.945 \\
SF-BSE@{\GOWO}@UHF & 2.399 & 6.191 & 7.792 & 9.373 \\
SF-BSE@{\evGW}@UHF & 2.407 & 6.199 & 7.788 & 9.388 \\
SF-BSE@{\qsGW}@UHF & 2.376 & 6.241 & 7.668 & 9.417 \\
SF-dBSE@{\GOWO}@UHF & 2.363 & 6.263 & 7.824 & 9.424 \\
SF-dBSE@{\evGW}@UHF & 2.369 & 6.273 & 7.820 & 9.441 \\
SF-dBSE@{\qsGW}@UHF & 2.335 & 6.317 & 7.689 & 9.470 \\
SF-ADC(2)-s & 2.433 & 6.255 & 7.745 & 9.047 \\
SF-ADC(2)-x & 2.866 & 6.581 & 7.664 & 8.612 \\
SF-ADC(3) & 2.863 & 6.579 & 7.658 & 8.618 \\
FCI & 2.862 & 6.577 & 7.669 & 8.624 \\
SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\
SF-TD-B3LYP\fnm[1] & 3.332 & 4.275 & 6.864 & 7.762 \\
SF-TD-BH\&HLYP\fnm[1] & 2.874 & 4.922 & 7.112 & 8.188 \\
SF-BSE@{\GOWO}\fnm[2] & 2.399 & 6.191 & 7.792 & 9.373 \\
SF-BSE@{\evGW}\fnm[2] & 2.407 & 6.199 & 7.788 & 9.388 \\
SF-BSE@{\qsGW}\fnm[2] & 2.376 & 6.241 & 7.668 & 9.417 \\
SF-dBSE@{\GOWO}\fnm[2] & 2.363 & 6.263 & 7.824 & 9.424 \\
SF-dBSE@{\evGW}\fnm[2] & 2.369 & 6.273 & 7.820 & 9.441 \\
SF-dBSE@{\qsGW}\fnm[2] & 2.335 & 6.317 & 7.689 & 9.470 \\
SF-CIS\fnm[3] & 2.111 & 6.036 & 7.480 & 8.945 \\
SF-ADC(2)-s\fnm[2] & 2.433 & 6.255 & 7.745 & 9.047 \\
SF-ADC(2)-x\fnm[2] & 2.866 & 6.581 & 7.664 & 8.612 \\
SF-ADC(3)\fnm[2] & 2.863 & 6.579 & 7.658 & 8.618 \\
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[2]{This work.}
\fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
%%% FIG. 1 %%%
\begin{figure}
\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).
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}}
\end{figure}
%%% TABLE II %%%
%\begin{squeezetable}

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