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@ -1,13 +1,41 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-12-09 22:57:40 +0100 %% Created for Pierre-Francois Loos at 2021-01-10 15:01:27 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Krylov_2000b,
author = {Krylov,Anna I.},
date-added = {2021-01-10 15:01:14 +0100},
date-modified = {2021-01-10 15:01:23 +0100},
doi = {10.1063/1.1308557},
eprint = {https://doi.org/10.1063/1.1308557},
journal = {The Journal of Chemical Physics},
number = {15},
pages = {6052-6062},
title = {Spin-contamination of coupled-cluster wave functions},
url = {https://doi.org/10.1063/1.1308557},
volume = {113},
year = {2000},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1308557}}
@article{Balkova_1994,
author = {Balkov{\'a},A. and Bartlett,Rodney J.},
date-added = {2021-01-10 14:55:51 +0100},
date-modified = {2021-01-10 14:56:05 +0100},
doi = {10.1063/1.468025},
journal = {J. Chem. Phys.},
number = {10},
pages = {8972-8987},
title = {A multireference coupled‐cluster study of the ground state and lowest excited states of cyclobutadiene},
volume = {101},
year = {1994},
Bdsk-Url-1 = {https://doi.org/10.1063/1.468025}}
@article{qchem4, @article{qchem4,
author = {Shao, Yihan and Gan, Zhengting and Epifanovsky, Evgeny and Gilbert, Andrew T.B. and Wormit, Michael and Kussmann, Joerg and Lange, Adrian W. and Behn, Andrew and Deng, Jia and Feng, Xintian and Ghosh, Debashree and Goldey, Matthew and Horn, Paul R. and Jacobson, Leif D. and Kaliman, Ilya and Khaliullin, Rustam Z. and Ku{\'s}, Tomasz and Landau, Arie and Liu, Jie and Proynov, Emil I. and Rhee, Young Min and Richard, Ryan M. and Rohrdanz, Mary A. and Steele, Ryan P. and Sundstrom, Eric J. and Woodcock, H. Lee and Zimmerman, Paul M. and Zuev, Dmitry and Albrecht, Ben and Alguire, Ethan and Austin, Brian and Beran, Gregory J. O. and Bernard, Yves A. and Berquist, Eric and Brandhorst, Kai and Bravaya, Ksenia B. and Brown, Shawn T. and Casanova, David and Chang, Chun-Min and Chen, Yunqing and Chien, Siu Hung and Closser, Kristina D. and Crittenden, Deborah L. and Diedenhofen, Michael and DiStasio, Robert A. and Do, Hainam and Dutoi, Anthony D. and Edgar, Richard G. and Fatehi, Shervin and Fusti-Molnar, Laszlo and Ghysels, An and Golubeva-Zadorozhnaya, Anna and Gomes, Joseph and Hanson-Heine, Magnus W.D. and Harbach, Philipp H.P. and Hauser, Andreas W. and Hohenstein, Edward G. and Holden, Zachary C. and Jagau, Thomas-C. and Ji, Hyunjun and Kaduk, Benjamin and Khistyaev, Kirill and Kim, Jaehoon and Kim, Jihan and King, Rollin A. and Klunzinger, Phil and Kosenkov, Dmytro and Kowalczyk, Tim and Krauter, Caroline M. and Lao, Ka Un and Laurent, Ad{\`e}le D. and Lawler, Keith V. and Levchenko, Sergey V. and Lin, Ching Yeh and Liu, Fenglai and Livshits, Ester and Lochan, Rohini C. and Luenser, Arne and Manohar, Prashant and Manzer, Samuel F. and Mao, Shan-Ping and Mardirossian, Narbe and Marenich, Aleksandr V. and Maurer, Simon A. and Mayhall, Nicholas J. and Neuscamman, Eric and Oana, C. Melania and Olivares-Amaya, Roberto and O'Neill, Darragh P. and Parkhill, John A. and Perrine, Trilisa M. and Peverati, Roberto and Prociuk, Alexander and Rehn, Dirk R. and Rosta, Edina and Russ, Nicholas J. and Sharada, Shaama M. and Sharma, Sandeep and Small, David W. and Sodt, Alexander and Stein, Tamar and St{\"u}ck, David and Su, Yu-Chuan and Thom, Alex J.W. and Tsuchimochi, Takashi and Vanovschi, Vitalii and Vogt, Leslie and Vydrov, Oleg and Wang, Tao and Watson, Mark A. and Wenzel, Jan and White, Alec and Williams, Christopher F. and Yang, Jun and Yeganeh, Sina and Yost, Shane R. and You, Zhi-Qiang and Zhang, Igor Ying and Zhang, Xing and Zhao, Yan and Brooks, Bernard R. and Chan, Garnet K.L. and Chipman, Daniel M. and Cramer, Christopher J. and Goddard, William A. and Gordon, Mark S. and Hehre, Warren J. and Klamt, Andreas and Schaefer, Henry F. and Schmidt, Michael W. and Sherrill, C. David and Truhlar, Donald G. and Warshel, Arieh and Xu, Xin and Aspuru-Guzik, Al{\'a}n and Baer, Roi and Bell, Alexis T. and Besley, Nicholas A. and Chai, Jeng-Da and Dreuw, Andreas and Dunietz, Barry D. and Furlani, Thomas R. and Gwaltney, Steven R. and Hsu, Chao-Ping and Jung, Yousung and Kong, Jing and Lambrecht, Daniel S. and Liang, WanZhen and Ochsenfeld, Christian and Rassolov, Vitaly A. and Slipchenko, Lyudmila V. and Subotnik, Joseph E. and Van Voorhis, Troy and Herbert, John M. and Krylov, Anna I. and Gill, Peter M.W. and Head-Gordon, Martin}, author = {Shao, Yihan and Gan, Zhengting and Epifanovsky, Evgeny and Gilbert, Andrew T.B. and Wormit, Michael and Kussmann, Joerg and Lange, Adrian W. and Behn, Andrew and Deng, Jia and Feng, Xintian and Ghosh, Debashree and Goldey, Matthew and Horn, Paul R. and Jacobson, Leif D. and Kaliman, Ilya and Khaliullin, Rustam Z. and Ku{\'s}, Tomasz and Landau, Arie and Liu, Jie and Proynov, Emil I. and Rhee, Young Min and Richard, Ryan M. and Rohrdanz, Mary A. and Steele, Ryan P. and Sundstrom, Eric J. and Woodcock, H. Lee and Zimmerman, Paul M. and Zuev, Dmitry and Albrecht, Ben and Alguire, Ethan and Austin, Brian and Beran, Gregory J. O. and Bernard, Yves A. and Berquist, Eric and Brandhorst, Kai and Bravaya, Ksenia B. and Brown, Shawn T. and Casanova, David and Chang, Chun-Min and Chen, Yunqing and Chien, Siu Hung and Closser, Kristina D. and Crittenden, Deborah L. and Diedenhofen, Michael and DiStasio, Robert A. and Do, Hainam and Dutoi, Anthony D. and Edgar, Richard G. and Fatehi, Shervin and Fusti-Molnar, Laszlo and Ghysels, An and Golubeva-Zadorozhnaya, Anna and Gomes, Joseph and Hanson-Heine, Magnus W.D. and Harbach, Philipp H.P. and Hauser, Andreas W. and Hohenstein, Edward G. and Holden, Zachary C. and Jagau, Thomas-C. and Ji, Hyunjun and Kaduk, Benjamin and Khistyaev, Kirill and Kim, Jaehoon and Kim, Jihan and King, Rollin A. and Klunzinger, Phil and Kosenkov, Dmytro and Kowalczyk, Tim and Krauter, Caroline M. and Lao, Ka Un and Laurent, Ad{\`e}le D. and Lawler, Keith V. and Levchenko, Sergey V. and Lin, Ching Yeh and Liu, Fenglai and Livshits, Ester and Lochan, Rohini C. and Luenser, Arne and Manohar, Prashant and Manzer, Samuel F. and Mao, Shan-Ping and Mardirossian, Narbe and Marenich, Aleksandr V. and Maurer, Simon A. and Mayhall, Nicholas J. and Neuscamman, Eric and Oana, C. Melania and Olivares-Amaya, Roberto and O'Neill, Darragh P. and Parkhill, John A. and Perrine, Trilisa M. and Peverati, Roberto and Prociuk, Alexander and Rehn, Dirk R. and Rosta, Edina and Russ, Nicholas J. and Sharada, Shaama M. and Sharma, Sandeep and Small, David W. and Sodt, Alexander and Stein, Tamar and St{\"u}ck, David and Su, Yu-Chuan and Thom, Alex J.W. and Tsuchimochi, Takashi and Vanovschi, Vitalii and Vogt, Leslie and Vydrov, Oleg and Wang, Tao and Watson, Mark A. and Wenzel, Jan and White, Alec and Williams, Christopher F. and Yang, Jun and Yeganeh, Sina and Yost, Shane R. and You, Zhi-Qiang and Zhang, Igor Ying and Zhang, Xing and Zhao, Yan and Brooks, Bernard R. and Chan, Garnet K.L. and Chipman, Daniel M. and Cramer, Christopher J. and Goddard, William A. and Gordon, Mark S. and Hehre, Warren J. and Klamt, Andreas and Schaefer, Henry F. and Schmidt, Michael W. and Sherrill, C. David and Truhlar, Donald G. and Warshel, Arieh and Xu, Xin and Aspuru-Guzik, Al{\'a}n and Baer, Roi and Bell, Alexis T. and Besley, Nicholas A. and Chai, Jeng-Da and Dreuw, Andreas and Dunietz, Barry D. and Furlani, Thomas R. and Gwaltney, Steven R. and Hsu, Chao-Ping and Jung, Yousung and Kong, Jing and Lambrecht, Daniel S. and Liang, WanZhen and Ochsenfeld, Christian and Rassolov, Vitaly A. and Slipchenko, Lyudmila V. and Subotnik, Joseph E. and Van Voorhis, Troy and Herbert, John M. and Krylov, Anna I. and Gill, Peter M.W. and Head-Gordon, Martin},
date-added = {2020-12-09 22:55:07 +0100}, date-added = {2020-12-09 22:55:07 +0100},
@ -23,7 +51,7 @@
@misc{g09, @misc{g09,
author = {M. J. Frisch and G. W. Trucks and H. B. Schlegel and G. E. Scuseria and M. A. Robb and J. R. Cheeseman and G. Scalmani and V. Barone and B. Mennucci and G. A. Petersson and H. Nakatsuji and M. Caricato and X. Li and H. P. Hratchian and A. F. Izmaylov and J. Bloino and G. Zheng and J. L. Sonnenberg and M. Hada and M. Ehara and K. Toyota and R. Fukuda and J. Hasegawa and M. Ishida and T. Nakajima and Y. Honda and O. Kitao and H. Nakai and T. Vreven and Montgomery, {Jr.}, J. A. and J. E. Peralta and F. Ogliaro and M. Bearpark and J. J. Heyd and E. Brothers and K. N. Kudin and V. N. Staroverov and R. Kobayashi and J. Normand and K. Raghavachari and A. Rendell and J. C. Burant and S. S. Iyengar and J. Tomasi and M. Cossi and N. Rega and J. M. Millam and M. Klene and J. E. Knox and J. B. Cross and V. Bakken and C. Adamo and J. Jaramillo and R. Gomperts and R. E. Stratmann and O. Yazyev and A. J. Austin and R. Cammi and C. Pomelli and J. W. Ochterski and R. L. Martin and K. Morokuma and V. G. Zakrzewski and G. A. Voth and P. Salvador and J. J. Dannenberg and S. Dapprich and A. D. Daniels and {\"O}. Farkas and J. B. Foresman and J. V. Ortiz and J. Cioslowski and D. J. Fox}, author = {M. J. Frisch and G. W. Trucks and H. B. Schlegel and G. E. Scuseria and M. A. Robb and J. R. Cheeseman and G. Scalmani and V. Barone and B. Mennucci and G. A. Petersson and H. Nakatsuji and M. Caricato and X. Li and H. P. Hratchian and A. F. Izmaylov and J. Bloino and G. Zheng and J. L. Sonnenberg and M. Hada and M. Ehara and K. Toyota and R. Fukuda and J. Hasegawa and M. Ishida and T. Nakajima and Y. Honda and O. Kitao and H. Nakai and T. Vreven and Montgomery, {Jr.}, J. A. and J. E. Peralta and F. Ogliaro and M. Bearpark and J. J. Heyd and E. Brothers and K. N. Kudin and V. N. Staroverov and R. Kobayashi and J. Normand and K. Raghavachari and A. Rendell and J. C. Burant and S. S. Iyengar and J. Tomasi and M. Cossi and N. Rega and J. M. Millam and M. Klene and J. E. Knox and J. B. Cross and V. Bakken and C. Adamo and J. Jaramillo and R. Gomperts and R. E. Stratmann and O. Yazyev and A. J. Austin and R. Cammi and C. Pomelli and J. W. Ochterski and R. L. Martin and K. Morokuma and V. G. Zakrzewski and G. A. Voth and P. Salvador and J. J. Dannenberg and S. Dapprich and A. D. Daniels and {\"O}. Farkas and J. B. Foresman and J. V. Ortiz and J. Cioslowski and D. J. Fox},
note = {Gaussian Inc. Wallingford CT 2009}, note = {Gaussian Inc. Wallingford CT 2009},
title = {Gaussian∼09 {R}evision {E}.01}} title = {Gaussin~09 {R}evision {E}.01}}
@article{QChem, @article{QChem,
author = {Shao, Y. and Fusti-Molnar, L. and Jung, Y. and Kussmann, J. and Ochsenfeld, C. and Brown, S. T. and Gilbert, A. T. B. and Slipchenko, L. V. and Levchenko, S. V. and O'Neill, D. P. and Distasio Jr., R. A. and Lochan, R. C. and Wang, T. and Beran, G. J. O. and Besley, N. A. and Herbert, J. M. and Lin, C. Y. and Van Voorhis, T. and Chien, S. H. and Sodt, A. and Steele, R. P. and Rassolov, V. A. and Maslen, P. E. and Korambath, P. P. and Adamson, R. D. and Austin, B. and Baker, J. and Byrd, E. F. C. and Dachsel, H. and Doerksen, R. J. and Dreuw, A. and Dunietz, B. D. and Dutoi, A. D. and Furlani, T. R. and Gwaltney, S. R. and Heyden, A. and Hirata, S. and Hsu, C.-P. and Kedziora, G. and Khalliulin, R. Z. and Klunzinger, P. and Lee, A. M. and Lee, M. S. and Liang, W. and Lotan, I. and Nair, N. and Peters, B. and Proynov, E. I. and Pieniazek, P. A. and Rhee, Y. M. and Ritchie, J. and Rosta, E. and Sherrill, C. D. and Simmonett, A. C. and Subotnik, J. E. and Woodcock III, H. L. and Zhang, W. and Bell, A. T. and Chakraborty, A. K. and Chipman, D. M. and Keil, F. J. and Warshel, A. and Hehre, W. J. and Schaefer III, H. F. and Kong , J. and Krylov, A. I. and Gill, P. M. W. and Head-Gordon, M.}, author = {Shao, Y. and Fusti-Molnar, L. and Jung, Y. and Kussmann, J. and Ochsenfeld, C. and Brown, S. T. and Gilbert, A. T. B. and Slipchenko, L. V. and Levchenko, S. V. and O'Neill, D. P. and Distasio Jr., R. A. and Lochan, R. C. and Wang, T. and Beran, G. J. O. and Besley, N. A. and Herbert, J. M. and Lin, C. Y. and Van Voorhis, T. and Chien, S. H. and Sodt, A. and Steele, R. P. and Rassolov, V. A. and Maslen, P. E. and Korambath, P. P. and Adamson, R. D. and Austin, B. and Baker, J. and Byrd, E. F. C. and Dachsel, H. and Doerksen, R. J. and Dreuw, A. and Dunietz, B. D. and Dutoi, A. D. and Furlani, T. R. and Gwaltney, S. R. and Heyden, A. and Hirata, S. and Hsu, C.-P. and Kedziora, G. and Khalliulin, R. Z. and Klunzinger, P. and Lee, A. M. and Lee, M. S. and Liang, W. and Lotan, I. and Nair, N. and Peters, B. and Proynov, E. I. and Pieniazek, P. A. and Rhee, Y. M. and Ritchie, J. and Rosta, E. and Sherrill, C. D. and Simmonett, A. C. and Subotnik, J. E. and Woodcock III, H. L. and Zhang, W. and Bell, A. T. and Chakraborty, A. K. and Chipman, D. M. and Keil, F. J. and Warshel, A. and Hehre, W. J. and Schaefer III, H. F. and Kong , J. and Krylov, A. I. and Gill, P. M. W. and Head-Gordon, M.},
@ -1550,10 +1578,10 @@
year = {1998}, year = {1998},
Bdsk-Url-1 = {https://doi.org/10.1063/1.477023}} Bdsk-Url-1 = {https://doi.org/10.1063/1.477023}}
@article{Krylov_2000, @article{Krylov_2000a,
author = {Krylov,Anna I. and Sherrill,C. David and Head-Gordon,Martin}, author = {Krylov,Anna I. and Sherrill,C. David and Head-Gordon,Martin},
date-added = {2020-12-06 14:34:27 +0100}, date-added = {2020-12-06 14:34:27 +0100},
date-modified = {2020-12-06 14:34:33 +0100}, date-modified = {2021-01-10 15:01:27 +0100},
doi = {10.1063/1.1311292}, doi = {10.1063/1.1311292},
eprint = {https://doi.org/10.1063/1.1311292}, eprint = {https://doi.org/10.1063/1.1311292},
journal = {The Journal of Chemical Physics}, journal = {The Journal of Chemical Physics},

214
Manuscript/sfBSE.tex
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@ -60,7 +60,7 @@ We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casa
Note that a similar idea has been exploited by the group of Weito Yang to access double excitations in the context of the particle-particle random-phase approximation. \cite{Peng_2013,Yang_2013b,Yang_2014a,Peng_2014,Zhang_2016,Sutton_2018} Note that a similar idea has been exploited by the group of Weito Yang to access double excitations in the context of the particle-particle random-phase approximation. \cite{Peng_2013,Yang_2013b,Yang_2014a,Peng_2014,Zhang_2016,Sutton_2018}
One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way. One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way.
Moreover, spin-flip methods are usually hampered by spin-contamination (\ie, artificial mixing with configurations of different spin multiplicities) due to spin incompleteness of the configuration interaction expansion as well as the possible spin-contamination of the reference configuration. Moreover, spin-flip methods are usually hampered by spin-contamination \cite{Casanova_2020} (\ie, artificial mixing with configurations of different spin multiplicities) due to spin incompleteness of the configuration interaction expansion as well as the possible spin-contamination of the reference configuration. \cite{Krylov_2000b}
This issue can be alleviated by increasing the excitation order at a significant cost or by selectively complementing the spin-incomplete configuration set with the missing configurations. \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018} This issue can be alleviated by increasing the excitation order at a significant cost or by selectively complementing the spin-incomplete configuration set with the missing configurations. \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
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. 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.
@ -70,7 +70,7 @@ The present BSE calculations are based on the spin unrestricted version of both
To the best of our knowledge, the present study is the first to apply the spin-flip formalism to the BSE method. To the best of our knowledge, the present study is the first to apply the spin-flip formalism to the BSE method.
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} 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}
We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}). We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}).
Computational details are reported in Sec.~\ref{sec:compdet} and our results for the beryllium atom \ce{Be}, the hydrogen molecule \ce{H2}, and cyclobutadiene \ce{C4H4} are discussed in Sec.~\ref{sec:res}. Computational details are reported in Sec.~\ref{sec:compdet} and our results for the beryllium atom \ce{Be} (Subsec.~\ref{sec:Be}), the hydrogen molecule \ce{H2} (Subsec.~\ref{sec:H2}), and cyclobutadiene \ce{C4H4} (Subsec.~\ref{sec:CBD}) are discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}. Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
Unless otherwise stated, atomic units are used. Unless otherwise stated, atomic units are used.
@ -585,6 +585,11 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
\label{sec:res} \label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===============================
\subsection{Beryllium atom}
\label{sec:Be}
%===============================
%%% TABLE I %%% %%% TABLE I %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
@ -610,100 +615,100 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
%%% TABLE II %%% %%% TABLE II %%%
\begin{squeezetable} %\begin{squeezetable}
\begin{table*} %\begin{table*}
\caption{ % \caption{
Spin-flip excitations (in eV) of \ce{H2} obtained for various methods with the cc-pVQZ basis. % Spin-flip excitations (in eV) of \ce{H2} obtained for various methods with the cc-pVQZ basis.
The $GW$ calculations are performed with a HF starting point. % The $GW$ calculations are performed with a HF starting point.
\label{tab:H2}} % \label{tab:H2}}
\begin{ruledtabular} %\begin{ruledtabular}
\begin{tabular}{lccccccccccccc}% seven columns now, not six... %\begin{tabular}{lccccccccccccc}% seven columns now, not six...
%
Distance (\AA) & \multicolumn{3}{c}{SF-CIS} & \multicolumn{3}{c}{SF-TDDFT} & \multicolumn{3}{c}{SF-BSE@{\GOWO}} & \multicolumn{3}{c}{EOM-CCSD}\\ \hline %Distance (\AA) & \multicolumn{3}{c}{SF-CIS} & \multicolumn{3}{c}{SF-TDDFT} & \multicolumn{3}{c}{SF-BSE@{\GOWO}} & \multicolumn{3}{c}{EOM-CCSD}\\ \hline
%
& $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ \\ % & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ & $B {}^1 \Sigma_u^+$ & $E {}^1 \Sigma_g^+$ & $F {}^1 \Sigma_g^+$ \\
%
\hline % \hline
%
0.50 & \\ %0.50 & \\
0.55 & \\ %0.55 & \\
0.60 & \\ %0.60 & \\
0.65 & \\ %0.65 & \\
0.70 & \\ %0.70 & \\
0.75 & \\ %0.75 & \\
0.80 & \\ %0.80 & \\
0.85 & \\ %0.85 & \\
0.90 & \\ %0.90 & \\
0.95 & \\ %0.95 & \\
1.00 & \\ %1.00 & \\
1.05 & \\ %1.05 & \\
1.10 & \\ %1.10 & \\
1.15 & \\ %1.15 & \\
1.20 & \\ %1.20 & \\
1.25 & \\ %1.25 & \\
1.30 & \\ %1.30 & \\
1.35 & \\ %1.35 & \\
1.40 & \\ %1.40 & \\
1.45 & \\ %1.45 & \\
1.50 & \\ %1.50 & \\
1.55 & \\ %1.55 & \\
1.60 & \\ %1.60 & \\
1.65 & \\ %1.65 & \\
1.70 & \\ %1.70 & \\
1.75 & \\ %1.75 & \\
1.80 & \\ %1.80 & \\
1.85 & \\ %1.85 & \\
1.90 & \\ %1.90 & \\
1.95 & \\ %1.95 & \\
2.00 & \\ %2.00 & \\
2.05 & \\ %2.05 & \\
2.10 & \\ %2.10 & \\
2.15 & \\ %2.15 & \\
2.20 & \\ %2.20 & \\
2.25 & \\ %2.25 & \\
2.30 & \\ %2.30 & \\
2.35 & \\ %2.35 & \\
2.40 & \\ %2.40 & \\
2.45 & \\ %2.45 & \\
2.50 & \\ %2.50 & \\
2.55 & \\ %2.55 & \\
2.60 & \\ %2.60 & \\
2.65 & \\ %2.65 & \\
2.70 & \\ %2.70 & \\
2.75 & \\ %2.75 & \\
2.80 & \\ %2.80 & \\
2.85 & \\ %2.85 & \\
2.90 & \\ %2.90 & \\
2.95 & \\ %2.95 & \\
3.00 & \\ %3.00 & \\
3.05 & \\ %3.05 & \\
3.10 & \\ %3.10 & \\
3.15 & \\ %3.15 & \\
3.20 & \\ %3.20 & \\
3.25 & \\ %3.25 & \\
3.30 & \\ %3.30 & \\
3.35 & \\ %3.35 & \\
3.40 & \\ %3.40 & \\
3.45 & \\ %3.45 & \\
3.50 & \\ %3.50 & \\
3.55 & \\ %3.55 & \\
3.60 & \\ %3.60 & \\
3.65 & \\ %3.65 & \\
3.70 & \\ %3.70 & \\
3.75 & \\ %3.75 & \\
3.80 & \\ %3.80 & \\
3.85 & \\ %3.85 & \\
3.90 & \\ %3.90 & \\
3.95 & \\ %3.95 & \\
4.00 & \\ %4.00 & \\
%
%
%
\end{tabular} %\end{tabular}
\end{ruledtabular} %\end{ruledtabular}
\end{table*} %\end{table*}
\end{squeezetable} %\end{squeezetable}
%%% %%% %%% %%% %%%% %%% %%% %%%
%%% FIG 1 %%% %%% FIG 1 %%%
@ -714,6 +719,23 @@ Distance (\AA) & \multicolumn{3}{c}{SF-CIS} & \multicolumn{3}{c}{SF-TDDFT} & \mu
%\label{fig:Be}} %\label{fig:Be}}
%\end{figure*} %\end{figure*}
%===============================
\subsection{Hydrogen molecule}
\label{sec:H2}
%===============================
%===============================
\subsection{Cyclobutadiene}
\label{sec:CBD}
%===============================
Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Manohar_2008,Lefrancois_2015}
%with potential large spin contamination.
In its $D_{2h}$ rectangular $^1 A_g$ ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals
However, in its $D_{4h}$ square-planar $^3 A_{2g}$ ground-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals.
In this case, single-reference methods notoriously fail.
Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:ccl} \label{sec:ccl}