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\section{Introduction}
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\section{Introduction}
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\label{sec:intro}
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\label{sec:intro}
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The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
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The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16} In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity that is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} \jt{Is it the screened Coulomb interaction which is responsible for these or more generally the Coulomb interaction? I would say the second.}
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The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
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The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
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{\GW} is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on plane waves \cite{Marini_2009, Deslippe_2012, Maggio_2017} or local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
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{\GW} is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on plane waves \cite{Marini_2009, Deslippe_2012, Maggio_2017} or local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
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