reformating Be table
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2021-01-10 15:01:27 +0100
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%% Created for Pierre-Francois Loos at 2021-01-10 18:27:42 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Sarkar_2021,
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author = {R. Sarkar and M. Boggio-Pasqua and P. F. Loos and D. Jacquemin},
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date-added = {2021-01-10 18:27:30 +0100},
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date-modified = {2021-01-10 18:27:30 +0100},
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journal = {J. Chem. Theory Comput.},
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title = {Benchmark of TD-DFT and Wavefunction Methods for Oscillator Strengths and Excited-State Dipoles},
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year = {submitted}}
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@article{Krylov_2000b,
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author = {Krylov,Anna I.},
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date-added = {2021-01-10 15:01:14 +0100},
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@ -1136,16 +1144,6 @@
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volume = {94},
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year = {2006}}
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@misc{Sarkar_2020,
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archiveprefix = {arXiv},
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author = {Rudraditya Sarkar and Martial Boggio-Pasqua and Pierre-Fran{\c c}ois Loos and Denis Jacquemin},
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date-added = {2020-12-09 09:59:26 +0100},
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date-modified = {2020-12-09 09:59:26 +0100},
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eprint = {2011.13233},
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primaryclass = {physics.chem-ph},
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title = {Benchmarking TD-DFT and Wave Function Methods for Oscillator Strengths and Excited-State Dipole Moments},
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year = {2020}}
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@article{Scemama_2018a,
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author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos},
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date-added = {2020-12-09 09:59:26 +0100},
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@ -65,7 +65,7 @@ This issue can be alleviated by increasing the excitation order at a significant
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Nowadays, spin-flip techniques are widely available for many types of methods such as equation-of-motion coupled cluster (EOM-CC), \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} configuration interaction (CI), \cite{Krylov_2001b,Krylov_2002,Mato_2018,Casanova_2008,Casanova_2009b} TD-DFT, \cite{Shao_2003,Wang_2004,Li_2011a,Bernard_2012,Zhang_2015} the algebraic-diagrammatic construction (ADC) scheme,\cite{Lefrancois_2015,Lefrancois_2016} and others \cite{Mayhall_2014a,Mayhall_2014b,Bell_2013,Mayhall_2014c} with successful applications in bond breaking processes, \cite{Golubeva_2007} radical chemistry, \cite{Slipchenko_2002,Wang_2005,Slipchenko_2003,Rinkevicius_2010,Ibeji_2015,Hossain_2017,Orms_2018,Luxon_2018} and photochemistry in general \cite{Casanova_2012,Gozem_2013,Nikiforov_2014,Lefrancois_2016} to mention a few.
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Here we apply the spin-flip technique to the BSE formalism in order to access, in particular, double excitations. \cite{Authier_2020}
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Here we apply the spin-flip technique to the BSE formalism in order to access, in particular, double excitations, \cite{Authier_2020} but not only.
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The present BSE calculations are based on the spin unrestricted version of both $GW$ (Sec.~\ref{sec:UGW}) and BSE (Sec.~\ref{sec:UBSE}).
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To the best of our knowledge, the present study is the first to apply the spin-flip formalism to the BSE method.
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Moreover, we also go beyond the static approximation by taking into account dynamical effects (Sec.~\ref{sec:dBSE}) via an unrestricted generalization of our recently developed (renormalized) perturbative correction which builds on the seminal work of Strinati, \cite{Strinati_1982,Strinati_1984,Strinati_1988} Romaniello and collaborators, \cite{Romaniello_2009b,Sangalli_2011} and Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Lettmann_2019}
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@ -526,7 +526,7 @@ where
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(p_\sig|x|q_\sigp) = \int \MO{p_\sig}(\br) \, x \, \MO{q_\sigp}(\br) d\br
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\end{equation}
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are one-electron integrals in the orbital basis.
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The total oscillator strength is given by
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The total oscillator strength in the so-called length gauge \cite{Chrayteh_2021,Sarkar_2021} is given by
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\begin{equation}
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f_{m}^{\spc} = \frac{2}{3} \Om{m}{\spc} \qty[ \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 ]
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\end{equation}
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@ -592,24 +592,34 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
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%%% TABLE I %%%
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\begin{squeezetable}
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\begin{table*}
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\begin{table}
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\caption{
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Spin-flip excitations (in eV) of \ce{Be} obtained for various methods with the 6-31G basis.
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The $GW$ calculations are performed with an UHF starting point.
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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.
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\label{tab:Be}}
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\begin{ruledtabular}
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\begin{tabular}{lcccccccccccc}
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State & TD-BLYP & TD-B3LYP & TD-BHHLYP & CIS
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& BSE@{\GOWO} & BSE@{\evGW} & dBSE@{\GOWO} & dBSE@{\evGW}
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& ADC(2) & ADC(2)-x & ADC(3) & FCI \\
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\begin{tabular}{lcccc}
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& \mc{4}{c}{Excitation energies (eV)} \\
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\cline{2-5}
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Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$
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& $^3P(1s^22p^2)$ & $^1P(1s^22p^2)$ \\
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\hline
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$^3P(1s^22s2p)$ & 3.210 & 3.332 & 2.874 & 2.111 & 2.399& 2.407& 2.363& 2.369 & 2.433 & 2.866 & 2.863 & 2.862 \\
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$^1P(1s^22s2p)$ & 3.210 & 4.275& 4.922& 6.036 & 6.191& 6.199 & 6.263 & 6.273 & 6.255 &6.581 & 6.579 & 6.577 \\
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$^3P(1s^22p^2)$ & 6.691 & 6.864& 7.112 & 7.480 & 7.792 & 7.788 & 7.824 & 7.820 & 7.745 & 7.664 & 7.658 & 7.669 \\
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$^1P(1s^22p^2)$ & 7.598 & 7.762& 8.188 & 8.945 &9.373 & 9.388 & 9.424 & 9.441 & 9.047 & 8.612 & 8.618 & 8.624 \\
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SF-TD-BLYP & 3.210 & 3.210 & 6.691 & 7.598 \\
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SF-TD-B3LYP & 3.332 & 4.275 & 6.864 & 7.762 \\
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SF-TD-BHHLYP & 2.874 & 4.922 & 7.112 & 8.188 \\
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SF-CIS & 2.111 & 6.036 & 7.480 & 8.945 \\
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SF-BSE@{\GOWO}@UHF & 2.399 & 6.191 & 7.792 & 9.373 \\
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SF-BSE@{\evGW}@UHF & 2.407 & 6.199 & 7.788 & 9.388 \\
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SF-BSE@{\qsGW}@UHF & 2.376 & 6.241 & 7.668 & 9.417 \\
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SF-dBSE@{\GOWO}@UHF & 2.363 & 6.263 & 7.824 & 9.424 \\
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SF-dBSE@{\evGW}@UHF & 2.369 & 6.273 & 7.820 & 9.441 \\
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SF-dBSE@{\qsGW}@UHF & 2.335 & 6.317 & 7.689 & 9.470 \\
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SF-ADC(2)-s & 2.433 & 6.255 & 7.745 & 9.047 \\
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SF-ADC(2)-x & 2.866 & 6.581 & 7.664 & 8.612 \\
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SF-ADC(3) & 2.863 & 6.579 & 7.658 & 8.618 \\
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FCI & 2.862 & 6.577 & 7.669 & 8.624 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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\end{table}
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\end{squeezetable}
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%%% %%% %%% %%%
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