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@ -301,7 +301,9 @@ Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V
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\varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
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\mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\GW}) - V^{\XC}}{\phi_p^{\KS}}.
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\end{equation}
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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}
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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.
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\titou{T2: Shall we introduce the renormalization factor?}
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This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988}
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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.
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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).
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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}
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@ -418,6 +420,27 @@ Then, these are used as input variables for the $GW$ calculation, whose main pur
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Depending on the level of self-consistency, only the eigenvalues or both the eigenvalues and the orbitals are updated.
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In the case of a $G_0W_0$ calculation, a single, perturbative correction is applied to the orbital energies only.
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The partially self-consistent ev$GW$ scheme update
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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}
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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}
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Improvements may be obtained via partial self-consistency while the conservation laws are satisfied at full self-consistency.
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However, things are not that simple, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
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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}
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In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.
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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}
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a paradigm central to many areas of physics and chemistry. \cite{Loos_2016}
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This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017}
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However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006}
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These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
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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}
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There exist two main types of partially self-consistent GW methods:
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i) \textit{``eigenvalue-only quasiparticle''} GW (ev$GW$), \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
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where the quasiparticle (QP) energies are updated at each iteration, and
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ii) \textit{``quasiparticle self-consistent''} GW (qs$GW$), \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011, Kaplan_2016}
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where one updates both the QP energies and the corresponding orbitals.
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Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
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\\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -626,6 +649,8 @@ Here goes the conclusion.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Acknowledgments}
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%%%%%%%%%%%%%%%%%%%%%%%%
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Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
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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''}.
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DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support.
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%%%%%%%%%%%%%%%%%%%%
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@ -664,6 +689,6 @@ and theoretical groups. He is the author of more than 500 scientific papers. He
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\begin{center}
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\includegraphics[width=3cm]{PFLoos}
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\end{center}
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\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.
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\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.
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\end{document}
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@ -1,13 +1,524 @@
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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 2020-05-15 09:00:18 +0200
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%% Created for Pierre-Francois Loos at 2020-05-15 15:13:22 +0200
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%% Saved with string encoding Unicode (UTF-8)
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|
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|
||||
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|
||||
Number = {15},
|
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|
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|
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|
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Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.52.11000}}
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|
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|
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Year = {2015},
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Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.92.081104}}
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Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.88.075105}}
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Author = {Caruso, Fabio},
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Date-Added = {2020-05-15 15:07:54 +0200},
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||||
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Title = {Self-Consistent {{GW}} Approach for the Unified Description of Ground and Excited States of Finite Systems},
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||||
Type = {{{PhD Thesis}}},
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||||
Year = {2013}}
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