From a257b714ccf4c15dde9d77551d78b2adf24f27ea Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 15 May 2020 15:13:52 +0200 Subject: [PATCH] refs --- Manuscript/BSE_JPCL.tex | 29 ++- Manuscript/xbbib.bib | 543 +++++++++++++++++++++++++++++++++++++--- 2 files changed, 539 insertions(+), 33 deletions(-) diff --git a/Manuscript/BSE_JPCL.tex b/Manuscript/BSE_JPCL.tex index 906784e..e95e01d 100644 --- a/Manuscript/BSE_JPCL.tex +++ b/Manuscript/BSE_JPCL.tex @@ -301,7 +301,9 @@ Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V \varepsilon_p^{\GW} = \varepsilon_p^{\KS} + \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\GW}) - V^{\XC}}{\phi_p^{\KS}}. \end{equation} -Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique. This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988} +Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique. +\titou{T2: Shall we introduce the renormalization factor?} +This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988} surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential. In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational scaling (see below). Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999} @@ -418,6 +420,27 @@ Then, these are used as input variables for the $GW$ calculation, whose main pur Depending on the level of self-consistency, only the eigenvalues or both the eigenvalues and the orbitals are updated. In the case of a $G_0W_0$ calculation, a single, perturbative correction is applied to the orbital energies only. The partially self-consistent ev$GW$ scheme update + +The simplest and most popular variant is perturbative GW, or $G_0W_0$, \cite{Hybertsen_1985, Hybertsen_1986} which has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018} +Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature, and violates some important conservation laws, such as the conservation of energy, momentum and particle number. \cite{Martin_1959, Baym_1961, Baym_1962} +Improvements may be obtained via partial self-consistency while the conservation laws are satisfied at full self-consistency. +However, things are not that simple, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook} +Indeed, there is a long-standing debate about the importance of partial and full self-consistency in GW. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018} +In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method. +A famous example has been provided by the calculations performed on the uniform electron gas, \cite{Holm_1998, Holm_1999,Holm_2000, Garcia-Gonzalez_2001} +a paradigm central to many areas of physics and chemistry. \cite{Loos_2016} +This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017} +However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006} +These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations. +For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent GW methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017} + + +There exist two main types of partially self-consistent GW methods: +i) \textit{``eigenvalue-only quasiparticle''} GW (ev$GW$), \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} +where the quasiparticle (QP) energies are updated at each iteration, and +ii) \textit{``quasiparticle self-consistent''} GW (qs$GW$), \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011, Kaplan_2016} +where one updates both the QP energies and the corresponding orbitals. +Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case. \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -626,6 +649,8 @@ Here goes the conclusion. %%%%%%%%%%%%%%%%%%%%%%%% \section*{Acknowledgments} %%%%%%%%%%%%%%%%%%%%%%%% +Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. +This work has also been supported through the EUR Grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}. DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support. %%%%%%%%%%%%%%%%%%%% @@ -664,6 +689,6 @@ and theoretical groups. He is the author of more than 500 scientific papers. He \begin{center} \includegraphics[width=3cm]{PFLoos} \end{center} -\noindent{\bfseries Pierre-Fran\c{c}ois Loos} was born in Nancy, France in 1982. He received his M.S.~in Computational and Theoretical Chemistry from the Universit\'e Henri Poincar\'e (Nancy, France) in 2005 and his Ph.D.~from the same university in 2008. From 2009 to 2013, He was undertaking postdoctoral research with Peter M.W.~Gill at the Australian National University (ANU). From 2013 to 2017, he was a \textit{``Discovery Early Career Researcher Award''} recipient at the ANU. Since 2017, he holds a researcher position from the \textit{``Centre National de la Recherche Scientifique (CNRS)} at the \textit{Laboratoire de Chimie et Physique Quantiques} in Toulouse (France), and was awarded, in 2019, an ERC consolidator grant for the development of new excited-state methodologies. +\noindent{\bfseries Pierre-Fran\c{c}ois Loos} was born in Nancy, France in 1982. He received his M.S.~in Computational and Theoretical Chemistry from the Universit\'e Henri Poincar\'e (Nancy, France) in 2005 and his Ph.D.~from the same university in 2008. From 2009 to 2013, He was undertaking postdoctoral research with Peter M.W.~Gill at the Australian National University (ANU). From 2013 to 2017, he was a \textit{``Discovery Early Career Researcher Award''} recipient and, then, a senior lecturer at the ANU. Since 2017, he holds a researcher position from the \textit{``Centre National de la Recherche Scientifique (CNRS)} at the \textit{Laboratoire de Chimie et Physique Quantiques} in Toulouse (France), and was awarded, in 2019, an ERC consolidator grant for the development of new excited-state methodologies. \end{document} diff --git a/Manuscript/xbbib.bib b/Manuscript/xbbib.bib index 6781c2f..5c46609 100644 --- a/Manuscript/xbbib.bib +++ b/Manuscript/xbbib.bib @@ -1,13 +1,524 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-05-15 09:00:18 +0200 +%% Created for Pierre-Francois Loos at 2020-05-15 15:13:22 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Martin_1959, + Author = {Martin, Paul C. and Schwinger, Julian}, + Date-Added = {2020-05-15 15:13:03 +0200}, + Date-Modified = {2020-05-15 15:13:03 +0200}, + Doi = {10.1103/PhysRev.115.1342}, + Issn = {0031-899X}, + Journal = {Phys. Rev.}, + Language = {en}, + Month = sep, + Number = {6}, + Pages = {1342--1373}, + Title = {Theory of {{Many}}-{{Particle Systems}}. {{I}}}, + Volume = {115}, + Year = {1959}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.115.1342}} + +@article{Friedrich_2006, + Author = {Friedrich, Christoph and Schindlmayr, Arno and Bl{\"u}gel, Stefan and Kotani, Takao}, + Date-Added = {2020-05-15 15:12:46 +0200}, + Date-Modified = {2020-05-15 15:12:46 +0200}, + Doi = {10.1103/PhysRevB.74.045104}, + Issn = {1098-0121, 1550-235X}, + Journal = {Phys. Rev. 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B}, + Language = {en}, + Month = aug, + Number = {8}, + Pages = {081104}, + Shorttitle = {Beyond the {{G W}} Approximation}, + Title = {Beyond the {{G W}} Approximation: {{A}} Second-Order Screened Exchange Correction}, + Volume = {92}, + Year = {2015}, + Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.92.081104}} + +@article{Caruso_2013, + Author = {Caruso, Fabio and Rohr, Daniel R. and Hellgren, Maria and Ren, Xinguo and Rinke, Patrick and Rubio, Angel and Scheffler, Matthias}, + Date-Added = {2020-05-15 15:07:54 +0200}, + Date-Modified = {2020-05-15 15:07:54 +0200}, + Doi = {10.1103/PhysRevLett.110.146403}, + File = {/Users/loos/Zotero/storage/UF9F4UWC/Caruso_2013a.pdf;/Users/loos/Zotero/storage/Y3IZUCIX/Caruso_2013b.pdf}, + Issn = {0031-9007, 1079-7114}, + Journal = {Phys. Rev. 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B}, + Language = {en}, + Month = aug, + Number = {7}, + Pages = {075105}, + Shorttitle = {Self-Consistent {{G W}}}, + Title = {Self-Consistent {{G W}} : {{All}}-Electron Implementation with Localized Basis Functions}, + Volume = {88}, + Year = {2013}, + Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.88.075105}} + +@phdthesis{Caruso_2013b, + Author = {Caruso, Fabio}, + Date-Added = {2020-05-15 15:07:54 +0200}, + Date-Modified = {2020-05-15 15:07:54 +0200}, + File = {/Users/loos/Zotero/storage/I5Q6ZQCS/Caruso_PhD.pdf}, + School = {Freie Universit{\"a}t Berlin}, + Title = {Self-Consistent {{GW}} Approach for the Unified Description of Ground and Excited States of Finite Systems}, + Type = {{{PhD Thesis}}}, + Year = {2013}} + +@article{Hellgren_2015, + Author = {Hellgren, Maria and Caruso, Fabio and Rohr, Daniel R. and Ren, Xinguo and Rubio, Angel and Scheffler, Matthias and Rinke, Patrick}, + Date-Added = {2020-05-15 15:07:54 +0200}, + Date-Modified = {2020-05-15 15:07:54 +0200}, + Doi = {10.1103/PhysRevB.91.165110}, + File = {/Users/loos/Zotero/storage/CXHEK6Z6/Hellgren_2015.pdf}, + Issn = {1098-0121, 1550-235X}, + Journal = {Phys. 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Lett.}, + Language = {en}, + Month = jul, + Number = {4}, + Pages = {788--791}, + Title = {Total {{Energies}} from {{GW Calculations}}}, + Volume = {83}, + Year = {1999}, + Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevLett.83.788}} + +@article{Holm_2000, + Author = {Holm, B. and Aryasetiawan, F.}, + Date-Added = {2020-05-15 15:07:36 +0200}, + Date-Modified = {2020-05-15 15:07:36 +0200}, + File = {/Users/loos/Zotero/storage/EX4CWKJL/Holm_2000.pdf}, + Journal = {Phys. Rev. B}, + Number = {8}, + Pages = {4858}, + Title = {Total Energy from the {{Galitskii}}-{{Migdal}} Formula Using Realistic Spectral Functions}, + Volume = {62}, + Year = {2000}} + @book{ParrBook, Address = {Clarendon Press}, Author = {R. G. Parr and W. Yang}, @@ -773,22 +1284,6 @@ Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1888}, Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.51.1888}} -@article{Hybertsen_1986, - Author = {Hybertsen, Mark S. and Louie, Steven G.}, - Doi = {10.1103/PhysRevB.34.5390}, - Issue = {8}, - Journal = {Phys. Rev. B}, - Month = {Oct}, - Numpages = {0}, - Pages = {5390--5413}, - Publisher = {American Physical Society}, - Title = {Electron Correlation in Semiconductors and Insulators: Band Gaps and Quasiparticle Energies}, - Url = {http://link.aps.org/doi/10.1103/PhysRevB.34.5390}, - Volume = {34}, - Year = {1986}, - Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevB.34.5390}, - Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.34.5390}} - @article{Godby_1988, Author = {Godby, R. W. and Schl\"uter, M. and Sham, L. J.}, Doi = {10.1103/PhysRevB.37.10159}, @@ -1858,20 +2353,6 @@ Year = {2016}, Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01238}} -@article{Caruso_2016, - Author = {Caruso, Fabio and Dauth, Matthias and van Setten, Michiel J. and Rinke, Patrick}, - Doi = {10.1021/acs.jctc.6b00774}, - Eprint = {https://doi.org/10.1021/acs.jctc.6b00774}, - Journal = {Journal of Chemical Theory and Computation}, - Note = {PMID: 27631585}, - Number = {10}, - Pages = {5076-5087}, - Title = {Benchmark of GW Approaches for the GW100 Test Set}, - Url = {https://doi.org/10.1021/acs.jctc.6b00774}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00774}} - @article{Korbel_2014, Author = {K{\"{o}}rbel, Sabine and Boulanger, Paul and Duchemin, Ivan and Blase, Xavier and Marques, Miguel A. L. and Botti, Silvana}, Doi = {10.1021/ct5003658},