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%% Created for Pierre-Francois Loos at 2020-12-09 14:38:47 +0100
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%% Created for Pierre-Francois Loos at 2020-12-09 22:57:40 +0100
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@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},
|
||||
date-added = {2020-12-09 22:55:07 +0100},
|
||||
date-modified = {2020-12-09 22:56:34 +0100},
|
||||
doi = {10.1080/00268976.2014.952696},
|
||||
journal = {Mol. Phys.},
|
||||
pages = {184-215},
|
||||
title = {Advances in Molecular Quantum Chemistry Contained in the Q-Chem 4 Program Package},
|
||||
volume = {113},
|
||||
year = {2015},
|
||||
Bdsk-Url-1 = {http://dx.doi.org/10.1080/00268976.2014.952696}}
|
||||
|
||||
@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},
|
||||
note = {Gaussian Inc. Wallingford CT 2009},
|
||||
title = {Gaussian∼09 {R}evision {E}.01}}
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||||
|
||||
@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.},
|
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date-added = {2020-12-09 22:47:45 +0100},
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date-modified = {2020-12-09 22:47:45 +0100},
|
||||
journal = PCCP,
|
||||
pages = {3172--3191},
|
||||
title = {Advances in methods and algorithms in a modern quantum chemistry program package},
|
||||
volume = {8},
|
||||
year = {2006}}
|
||||
|
||||
@article{Hirata_2004,
|
||||
author = {Hirata, S.},
|
||||
date-added = {2020-12-09 21:02:23 +0100},
|
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date-modified = {2020-12-09 21:02:23 +0100},
|
||||
journal = {J. Chem. Phys.},
|
||||
pages = {51--59},
|
||||
title = {Higher-Order Equation-of-Motion Coupled-Cluster Methods},
|
||||
volume = 121,
|
||||
year = 2004}
|
||||
|
||||
@article{Agboola_2015,
|
||||
author = {Agboola, Davids and Knol, Anneke L. and Gill, Peter M. W. and Loos, Pierre-Fran{\c c}ois},
|
||||
date-added = {2020-12-09 09:59:26 +0100},
|
||||
@ -11929,11 +11966,11 @@
|
||||
year = {2015},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b00022}}
|
||||
|
||||
@book{Ulrich_2012,
|
||||
@book{UlrichBook,
|
||||
address = {New York},
|
||||
author = {Ullrich, C.},
|
||||
date-added = {2020-01-01 21:36:51 +0100},
|
||||
date-modified = {2020-01-01 21:36:52 +0100},
|
||||
date-modified = {2020-12-09 21:41:58 +0100},
|
||||
publisher = {Oxford University Press},
|
||||
series = {Oxford Graduate Texts},
|
||||
title = {Time-Dependent Density-Functional Theory: Concepts and Applications},
|
||||
|
||||
@ -25,10 +25,10 @@
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||||
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||||
\begin{abstract}
|
||||
Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe-Salpeter equation (BSE) formalism in its static approximation is ``blind'' to double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes.
|
||||
Here, we apply the spin-flip technique to the BSE formalism of many-body perturbation theory in order to access double excitations.
|
||||
Here, we apply the spin-flip technique (which consists in considering the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism of many-body perturbation theory in order to access double excitations.
|
||||
The present scheme is based on a spin-unrestricted version of the $GW$ approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations.
|
||||
Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment.
|
||||
The performance of the present spin-flip BSE formalism is illustrated by computing the vertical excitation energies of the beryllium atom, the hydrogen molecule, and cyclobutadiene.
|
||||
The performance of the present spin-flip BSE formalism is illustrated by computing the vertical excitation energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene.
|
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%\bigskip
|
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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@ -48,38 +48,38 @@ Accurately predicting ground- and excited-state energies (hence excitation energ
|
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An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020d,Casanova_2020}
|
||||
The fact that none of these methods is successful in every chemical scenario has encouraged chemists to carry on the development of new excited-state methodologies, their main goal being to get the most accurate excitation energies (and properties) at the lowest possible computational cost in the most general context. \cite{Loos_2020d}
|
||||
|
||||
Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} and popularized in condensed-matter physics, \cite{Sham_1966,Strinati_1984,Delerue_2000} one of the new emerging method in the computational chemistry landscape is the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999,Blase_2018,Blase_2020} from many-body perturbation theory (MBPT) \cite{Onida_2002,Martin_2016} which, based on an underlying $GW$ calculation to compute accurate charged excitations, \cite{Hedin_1965,Golze_2019} is able to provide accurate optical (\ie, neutral) excitations for molecular systems at a rather modest computational cost.\cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020}
|
||||
Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} and popularized in condensed-matter physics, \cite{Sham_1966,Strinati_1984,Delerue_2000} one of the new emerging method in the computational chemistry landscape is the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999,Blase_2018,Blase_2020} from many-body perturbation theory (MBPT) \cite{Onida_2002,Martin_2016} which, based on an underlying $GW$ calculation to compute accurate charged excitations and the dynamically-screened Coulomb potential, \cite{Hedin_1965,Golze_2019} is able to provide accurate optical (\ie, neutral) excitations for molecular systems at a rather modest computational cost.\cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020}
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Most of BSE implementations rely on the so-called static approximation, \cite{Blase_2018,Bruneval_2016,Krause_2017,Liu_2020} which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
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||||
Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996} the static BSE formalism is plagued by the lack of double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
|
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Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996,UlrichBook} the static BSE formalism is plagued by the lack of double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes\cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019} or the ground state of open-shell molecules, \cite{Casida_2005,Huix-Rotllant_2011,Loos_2020f} and can be a real challenge to accurately predict even with state-of-the-art methods, \cite{Loos_2018a,Loos_2019,Loos_2020c,Loos_2020d,Veril_2020} like the approximate third-order coupled-cluster (CC3) method \cite{Christiansen_1995b,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
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Indeed, both adiabatic TD-DFT \cite{Levine_2006,Tozer_2000,Elliott_2011,Maitra_2012,Maitra_2016} and static BSE \cite{ReiningBook,Romaniello_2009b,Sangalli_2011,Loos_2020h,Authier_2020} can only access (singlet and triplet) single excitations with respect to the reference determinant usually taken as the closed-shell singlet ground state.
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One way to access double excitations is via the spin-flip formalism established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002} with earlier attempts by Bethe, \cite{Bethe_1931} as well as Shibuya and McKoy. \cite{Shibuya_1970}
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The idea behind spin-flip is rather simple: instead of considering the singlet ground state as reference, the reference is taken as the lowest triplet state.
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In such a way, one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation.
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We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a detailed review of spin-flip methods.
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We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for detailed reviews on spin-flip methods.
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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}
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One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way.
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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.
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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}
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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 the photochemistry of conical intersections \cite{Casanova_2012,Gozem_2013,Nikiforov_2014,Lefrancois_2016} to mention a few.
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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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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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The computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}) is also discussed.
|
||||
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}.
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
|
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Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
|
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Unless otherwise stated, atomic units are used.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Unrestricted $GW$ formalism}
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\label{sec:UGW}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons, respectively) and $N$ one-electron basis functions.
|
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The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming no linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
|
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The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming the absence of linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
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The number of spin-conserved (sc) single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
|
||||
Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
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It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$.
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@ -100,7 +100,7 @@ The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBo
|
||||
\end{equation}
|
||||
where $\eta$ is a positive infinitesimal.
|
||||
As readily seen in Eq.~\eqref{eq:G}, the Green's function can be evaluated at different levels of theory depending on the choice of orbitals and energies, $\MO{p_\sig}$ and $\e{p_\sig}{}$.
|
||||
For example, $G_{\KS}^{\sig}$ is the independent-particle Green's function built with KS orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$.
|
||||
For example, $G_{\KS}^{\sig}$ is the independent-particle Green's function built with KS orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$. \cite{Hohenberg_1964,Kohn_1965,ParrBook}
|
||||
Within self-consistent schemes, these quantities can be replaced by quasiparticle energies and orbitals evaluated within the $GW$ approximation (see below). \cite{Hedin_1965,Golze_2019}
|
||||
|
||||
Based on the spin-up and spin-down components of $G$ defined in Eq.~\eqref{eq:G}, one can easily compute the non-interacting polarizability (which is a sum over spins)
|
||||
@ -292,6 +292,7 @@ where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and it
|
||||
\end{equation}
|
||||
Because, from a practical point of view, one is usually interested by the so-called quasiparticle solution (or peak), the quasiparticle equation \eqref{eq:QP-eq} is often linearized around $\omega = \e{p_\sig}^{\KS}$, yielding
|
||||
\begin{equation}
|
||||
\label{eq:G0W0_lin}
|
||||
\eGW{p_\sig}
|
||||
= \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ]
|
||||
\end{equation}
|
||||
@ -301,11 +302,11 @@ where
|
||||
Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1}
|
||||
\end{equation}
|
||||
is a renormalization factor (with $0 \le Z_{p_\sig} \le 1$) which also represents the spectral weight of the quasiparticle solution.
|
||||
In addition to the principal quasiparticle peak which, in a well-behaved case, contains most of the spectral weight, the frequency-dependent quasiparticle equation \eqref{eq:QP-eq} generates a finite number of satellite resonances with smaller weights.
|
||||
In addition to the principal quasiparticle peak which, in a well-behaved case, contains most of the spectral weight, the frequency-dependent quasiparticle equation \eqref{eq:QP-eq} generates a finite number of satellite resonances with smaller weights. \cite{Loos_2018b}
|
||||
|
||||
Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the KS level).
|
||||
Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018} several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the KS level).
|
||||
|
||||
Finally, within the quasiparticle self-consistent $GW$ (qs$GW$) scheme, both the one-electron energies and the orbitals are updated until convergence is reached.
|
||||
Finally, within the quasiparticle self-consistent $GW$ (qs$GW$) scheme, \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} both the one-electron energies and the orbitals are updated until convergence is reached.
|
||||
These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and hermitian self-energy defined as
|
||||
\begin{equation}
|
||||
\Tilde{\Sigma}_{p_\sig q_\sig}^{\xc} = \frac{1}{2} \qty[ \Sig{p_\sig q_\sig}{\xc}(\e{p_\sig}{}) + \Sig{q_\sig p_\sig}{\xc}(\e{p_\sig}{}) ]
|
||||
@ -535,8 +536,8 @@ For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix e
|
||||
\subsection{Spin contamination}
|
||||
\label{sec:spin}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
One of the key issues of linear response formalism based on unrestricted references is spin contamination.
|
||||
As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval{\hS^2} > 2$ for a high-spin triplets, and ii) spin-contamination of the excited states due to spin incompleteness of the configuration interaction expansion.
|
||||
One of the key issues of linear response formalism based on unrestricted references is spin contamination or the artificial mixing with configurations of different spin multiplicities.
|
||||
As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval{\hS^2} > 2$ for a high-spin triplets, and ii) spin-contamination of the excited states due to spin incompleteness of the CI expansion.
|
||||
The latter issue is an important source of spin contamination in the present context as BSE is limited to single excitations with respect to the reference configuration.
|
||||
Specific schemes have been developed to palliate these shortcomings and we refer the interested reader to Ref.~\onlinecite{Casanova_2020} for a detailed discussion on this matter.
|
||||
|
||||
@ -557,29 +558,27 @@ is the expectation value of $\hS^2$ for the reference configuration, the first t
|
||||
\end{equation}
|
||||
are overlap integrals between spin-up and spin-down orbitals.
|
||||
|
||||
For a given single excitation $m$, the explicit expressions of $\Delta \expval{\hS^2}_m^{\spc}$ and $\Delta \expval{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the $\bX{m}{}$ and $\bY{m}{}$ vectors and the orbital overlaps defined in Eq.~\eqref{eq:OV}.
|
||||
For a given single excitation $m$, the explicit expressions of $\Delta \expval{\hS^2}_m^{\spc}$ and $\Delta \expval{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the vectors $\bX{m}{}$ and $\bY{m}{}$ as well as the orbital overlaps defined in Eq.~\eqref{eq:OV}.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational details}
|
||||
\label{sec:compdet}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
For the systems under investigation here, we consider either an open-shell doublet or triplet reference state.
|
||||
For the closed-shell systems under investigation here, we consider a triplet reference state.
|
||||
We then adopt the unrestricted formalism throughout this work.
|
||||
The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (unrestricted) UHF starting point.
|
||||
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a,Hybertsen_1986,vanSetten_2013} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
|
||||
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
|
||||
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
|
||||
%Note that, for the present (small) molecular systems, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies and fundamental gap.
|
||||
%Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
|
||||
%In the present study, the zeroth-order Hamiltonian [see Eq.~\eqref{eq:LR-PT}] is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
|
||||
The dynamical correction is computed in the TDA throughout.
|
||||
As one-electron basis sets, we employ the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.
|
||||
The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point.
|
||||
For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
|
||||
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}
|
||||
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.}
|
||||
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
|
||||
%It is important to mention that the small molecular systems considered here are particularly challenging for the BSE formalism, \cite{Hirose_2015,Loos_2018b} which is known to work best for larger systems where the amount of screening is more important. \cite{Jacquemin_2017b,Rangel_2017}
|
||||
|
||||
%For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted.
|
||||
%Various statistical quantities are reported in the following: the mean signed error (MSE), mean absolute error (MAE), root-mean-square error (RMSE), and the maximum positive [Max($+$)] and maximum negative [Max($-$)] errors.
|
||||
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}.
|
||||
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}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Results}
|
||||
|
||||
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