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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}}

% methods
\newcommand{\evGW}{evGW}	
\newcommand{\qsGW}{qsGW}	
\newcommand{\GOWO}{G$_0$W$_0$}	
\newcommand{\GW}{GW}		
\newcommand{\GnWn}[1]{G$_{#1}$W$_{#1}$}		

% operators
\newcommand{\hH}{\Hat{H}}

% energies
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E_\text{HF}}
\newcommand{\EKS}{E_\text{KS}}
\newcommand{\EcK}{E_\text{c}^\text{Klein}}
\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
\newcommand{\EcGM}{E_\text{c}^\text{GM}}
\newcommand{\EcMP}{E_c^\text{MP2}}
\newcommand{\Egap}{E_\text{gap}}
\newcommand{\IP}{\text{IP}}
\newcommand{\EA}{\text{EA}}
\newcommand{\RH}{R_{\ce{H2}}}
\newcommand{\RF}{R_{\ce{F2}}}
\newcommand{\RBeO}{R_{\ce{BeO}}}

% orbital energies
\newcommand{\nDIIS}{N^\text{DIIS}}
\newcommand{\maxDIIS}{N_\text{max}^\text{DIIS}}
\newcommand{\nSat}[1]{N_{#1}^\text{sat}}
\newcommand{\eSat}[2]{\epsilon_{#1,#2}}
\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
\newcommand{\de}[1]{\Delta\epsilon_{#1}}
\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
\newcommand{\Om}[1]{\Omega_{#1}}
\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
\newcommand{\eLUMO}{\epsilon_\text{LUMO}}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}

% Matrix elements
\newcommand{\A}[1]{A_{#1}}
\newcommand{\B}[1]{B_{#1}}
\newcommand{\tA}{\Tilde{A}}
\newcommand{\tB}{\Tilde{B}}
\renewcommand{\S}[1]{S_{#1}}
\newcommand{\G}[1]{G_{#1}}
\newcommand{\Po}[1]{P_{#1}}
\newcommand{\W}[1]{W_{#1}}
\newcommand{\Wc}[1]{W^\text{c}_{#1}}
\newcommand{\vc}[1]{v_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
\newcommand{\Z}[1]{Z_{#1}}

% Matrices
\newcommand{\bG}{\boldsymbol{G}}
\newcommand{\bW}{\boldsymbol{W}}
\newcommand{\bvc}{\boldsymbol{v}}
\newcommand{\bSig}{\boldsymbol{\Sigma}}
\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}}
\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}}
\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}}
\newcommand{\be}{\boldsymbol{\epsilon}}
\newcommand{\bDelta}{\boldsymbol{\Delta}}
\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}}
\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}}
\newcommand{\beGnWn}[1]{\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
\newcommand{\bdeGnWn}[1]{\Delta\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
\newcommand{\bde}{\boldsymbol{\Delta\epsilon}}
\newcommand{\bdeHF}{\boldsymbol{\Delta\epsilon}^\text{HF}}
\newcommand{\bdeGW}{\boldsymbol{\Delta\epsilon}^\text{GW}}
\newcommand{\bOm}{\boldsymbol{\Omega}}
\newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bB}{\boldsymbol{B}}
\newcommand{\bX}{\boldsymbol{X}}
\newcommand{\bY}{\boldsymbol{Y}}
\newcommand{\bZ}{\boldsymbol{Z}}

\newcommand{\fc}{f_\text{c}}
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\newcommand{\MO}[1]{\phi_{#1}}

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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}

\begin{document}	

\title{A Density-Based Basis Set Correction for GW Methods}

\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\affiliation{\LCT}
\author{Julien Toulouse}
\email[Corresponding author: ]{toulouse@lct.jussieu.fr}
\affiliation{\LCT}

\begin{abstract}
%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
%	\centering
%  \includegraphics[width=\linewidth]{TOC}
%\end{wrapfigure}
Similarly to other electron correlation methods, many-body perturbation theory methods, such as the so-called GW approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis functions due to the lack of explicit electron-electron terms modeling the infamous electron-electron cusp.
Here, we propose a density-based basis set correction which significantly speed up the convergence of energetics towards the complete basis set limit.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Many-body Green-function theory with DFT basis-set correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $N$-electron system with nuclei-electron potential $v_\text{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ which are ``representable'' in a finite basis set ${\cal B}$
\begin{equation}
E_0^{\cal B} = \min_{n \in {\cal D}^{\cal B}} \left\{ F[n] + \int v_\text{ne}(\b{r}) n(\b{r}) \d\b{r}\right\},
\label{E0B}
\end{equation}
where ${\cal D}^{\cal B}$ is the set of $N$-representable densities which can be extracted from a wave function $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$. In this expression, $F[n]=\min_{\Psi\to n} \bra{\Psi} \hat{T} + \hat{W}_\text{ee}\ket{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hat{T}$ and $\hat{W}_\text{ee}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
\begin{equation}
F[n] = F^{\cal B}[n] + \bar{E}^{\cal B}[n],
\label{Fn}
\end{equation}
where $F^{\cal B}[n]$ is the Levy-Lieb density functional with wave functions $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$
\begin{equation}
F^{\cal B}[n] = \min_{\Psi^{\cal B}\to n} \bra{\Psi^{\cal B}} \hat{T} + \hat{W}_\text{ee}\ket{\Psi^{\cal B}},
\end{equation}
and $\bar{E}^{\cal B}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^{\cal B}[n]$, we reexpress it with a contrained search over $N$-representable one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
\begin{equation}
F^{\cal B}[n] = \min_{G^{\cal B}\to n} \Omega^{\cal B}[G^{\cal B}],
\label{FBn}
\end{equation}
where $\Omega^{\cal B}[G]$ is chosen to be a Klein-like energy functional of the Green function (see, e.g., Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
\begin{equation}
\Omega^{\cal B}[G] = \Tr \left[\ln ( - G ) \right] - \Tr \left[ (G_\text{s}^{\cal B})^{-1} G -1 \right]  + \Phi_\text{Hxc}^{\cal B}[G],
\label{OmegaB}
\end{equation}
where $(G_\text{s}^{\cal B})^{-1}$ is the projection into ${\cal B}$ of the inverse free-particle Green function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint  \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^{\cal B}[G]$ is a Hartree-exchange-correlation (Hxc) functional of the Green functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\text{Hxc}^{\cal B}[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^{\cal B}[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
\begin{equation}
E_0^{\cal B} = \min_{G^{\cal B}} \left\{ \Omega^{\cal B}[G^{\cal B}] + \int v_\text{ne}(\b{r}) n_{G^{\cal B}}(\b{r}) \d\b{r} + \bar{E}^{\cal B}[n_{G^{\cal B}}] \right\},
\label{E0BGB}
\end{equation}
where the minimization is over $N$-representable one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$.

The stationary condition from Eq.~(\ref{E0BGB}) gives the following Dyson equation
\begin{equation}
(G^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1}- \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
\label{Dyson}
\end{equation}
where $(G_\text{0}^{\cal B})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $v_\text{ne}(\b{r})$,
$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^{\cal B}$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
\begin{equation}
\bar{\Sigma}^{\cal B}[n](\b{r},\b{r}') = \bar{v}^{\cal B}[n](\b{r}) \delta(\b{r}-\b{r}'),
\end{equation}
with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.

%From Julien:
%\begin{equation}
%\fdv{E[n_G]}{G(r,r',\omega)} = \int \fdv{E[n_G]}{n(r'')}] \fdv{n_G(r'')}{G(r,r',w)} dr''
%\end{equation}
%
%\begin{equation}
%n_G(r'')  = i \int  G(r'',r'',w) d\omega
%\end{equation}
%
%
%\begin{equation}
%\fdv{n_G(r'')}{G(r,r',w)} = \delta(r -r') \delta (r'-r'')
%\end{equation}
%
%
%\begin{equation}
%\begin{split}
%	\fdv{E[n_G]}{G(r,r',w)} 
%	& = \int \fdv{E[n_G]}{n(r'')}  \delta(r -r') \delta (r'-r'')  dr''
%	\\
%	& =  \fdv{E[n_G]}{n(r)} \delta(r -r')
%	\\
%	& =  v[n_G](r)  \delta(r -r')
%\end{split}
%\end{equation}

\subsection{The GW Approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Dyson equation can be written with an arbitrary reference
\begin{equation}
(G^{\cal B})^{-1} = (G_\text{ref}^{\cal B})^{-1}- \left( \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \Sigma_\text{ref}^{\cal B} \right)  - \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
\end{equation}
where $(G_\text{ref}^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1} - \Sigma_\text{ref}^{\cal B}$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^{\cal B}(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = v_\text{Hxc}^{\cal B}(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.


Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.

For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy is conveniently split in its hole (h) and particle (p) contributions
\begin{equation}
\label{eq:SigC}
	\SigC{p}(\omega) = \SigCp{p}(\omega) + \SigCh{p}(\omega),
\end{equation}
which, within the GW approximation, read
\begin{subequations}
\begin{align}
\label{eq:SigCh}
	\SigCh{p}(\omega)
	& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta},
	\\
\label{eq:SigCp}
	\SigCp{p}(\omega)
	& = 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
\end{align}
\end{subequations}
where $\eta$ is a positive infinitesimal.
The screened two-electron integrals
\begin{equation}
	[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x}
\end{equation}
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} $(pq|rs)$ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
\begin{equation}
\label{eq:LR}
	\begin{pmatrix}
		\bA	&	\bB	\\
		\bB	&	\bA	\\
	\end{pmatrix}
	\begin{pmatrix}
		\bX	\\
		\bY	\\
	\end{pmatrix}
	=
	\bOm
	\begin{pmatrix}
		\boldsymbol{1}	&	\boldsymbol{0}	\\
		\boldsymbol{0}	&	\boldsymbol{-1}	\\
	\end{pmatrix}
	\begin{pmatrix}
		\bX	\\
		\bY	\\
	\end{pmatrix},
\end{equation}
with
\begin{align}
\label{eq:RPA}
	A_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb),
	&
	B_{ia,jb} & = 2 (ia|bj),
\end{align}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
The one-electron energies $\epsilon_p$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} are either the HF or the GW quasiparticle energies.
Equation \eqref{eq:LR} also provides the neutral excitation energies $\Om{x}$.

In practice, there exist two ways of determining the {\GOWO} QP energies. \cite{Hybertsen_1985a, vanSetten_2013}
In its ``graphical'' version, they are provided by one of the many solutions of the (non-linear) QP equation
\begin{equation}
\label{eq:QP-G0W0}
	\omega =  \eHF{p} + \Re[\SigC{p}(\omega)].
\end{equation}
In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution.
In particular, it is usually worth calculating its renormalization weight (or factor), $\Z{p}(\eHF{p})$, where
\begin{equation}
\label{eq:Z}
	\Z{p}(\omega) = \qty[ 1 - \pdv{\Re[\SigC{p}(\omega)]}{\omega} ]^{-1}.
\end{equation}
Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
In a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.

Within the linearized version of {\GOWO}, one assumes that
\begin{equation}
	\label{eq:SigC-lin}
	\SigC{p}(\omega) \approx \SigC{p}(\eHF{p}) + (\omega - \eHF{p}) \left. \pdv{\SigC{p}(\omega)}{\omega} \right|_{\omega = \eHF{p}},
\end{equation}
that is, the self-energy behaves linearly in the vicinity of $\omega = \eHF{p}$.
Substituting \eqref{eq:SigC-lin} into \eqref{eq:QP-G0W0} yields
\begin{equation}
\label{eq:QP-G0W0-lin}
	\eGOWO{p} = \eHF{p} + \Z{p}(\eHF{p}) \Re[\SigC{p}(\eHF{p})].
\end{equation}
Unless otherwise stated, in the remaining of this paper, the {\GOWO} QP energies are determined via the linearized method.

In the case of {\evGW}, the QP energy, $\eGW{p}$, are obtained via Eq.~\eqref{eq:QP-G0W0}, which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref{eq:SigC}]. \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} 
At least in the weakly correlated regime where a clear QP solution exists, we believe that, within {\evGW}, the self-consistent algorithm should select the solution of the QP equation \eqref{eq:QP-G0W0} with the largest renormalization weight $\Z{p}(\eGW{p})$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Basis Set Correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The present basis set correction is a two-level correction.
%First, one has to correct the neutral excitations $\Om{x}$ from the RPA calculation.
%The corrected matrix elements read 
%\begin{align}
%\label{eq:RPA}
%	\tA_{ia,jb} & = \A{ia,jb} + (ia|\fc|jb),
%	&
%	\tB_{ia,jb} & = \B{ia,jb} + (ia|\fc|bj),
%\end{align}
%where the elements $\A{ia,jb}$ and $\B{ia,jb}$ are given by Eq.~\eqref{eq:RPA}.
%\begin{equation}
%	\fc(\br{1},\br{2})= \frac{\delta^2 \Ec}{\delta n(\br{1})\delta n(\br{2})}
%\end{equation}
%In a second time, we correct the GW energy
%\begin{equation}
%	\tSigC{p} = \SigC{p} + (p|\Vc|p)
%\end{equation}
%with
%\begin{equation}
%	\Vc(\br{}) = \fdv{\Ec}{n(\br{})}
%\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%

%%% TABLE I %%%
\begin{table*}
\caption{
IPs (in eV) of the five canonical nucleobases computed at various levels of theory.
\label{tab:DNA}
}
	\begin{ruledtabular}
		\begin{tabular}{llddddd}
									&				&	\mc{5}{c}{IPs of nucleobases (eV)}					\\	
														\cline{3-7}										
		Method						&	Basis		&	\tabc{Adenine} 	&	\tabc{Cytosine}	&	\tabc{Thymine}		&	\tabc{Guanine}	& \tabc{Uracil}	\\	
		\hline
		{\GOWO}@PBE\fnm[1]			&	def2-SVP	&	7.27	&	7.53	&	6.95	&	8.02	&	8.38	\\
		{\GOWO}@PBE+srLDA\fnm[1]	&	def2-SVP	&	7.60	&	7.95	&	7.29	&	8.36	&	8.80	\\
		{\GOWO}@PBE+srPBE\fnm[1]	&	def2-SVP	&	7.64	&	8.06	&	7.34	&	8.41	&	8.91	\\
		{\GOWO}@PBE\fnm[2]			&	def2-TZVP	&	7.75	&	8.07	&	7.46	&	8.49	&	9.02	\\
		{\GOWO}@PBE+srLDA\fnm[1]	&	def2-TZVP	&		&		&		&		&		\\
		{\GOWO}@PBE+srPBE\fnm[1]	&	def2-TZVP	&		&		&		&		&		\\
		{\GOWO}@PBE\fnm[2]			&	def2-QZVP	&	7.98	&	8.29	&	7.69	&	8.71	&	9.22	\\
		{\GOWO}@PBE\fnm[3]			&	def2-TQZVP	&	8.15	&	8.45	&	7.87	&	8.87	&	9.38	\\
		\hline
		CCSD(T)\fnm[4]				&	def2-TZVPP	&	8.33	&	9.51	&	8.03	&	9.08	&	10.13	\\
		Experiment\fnm[5]			&				&	8.48	&	8.94	&	8.24	&	9.2		&	9.68	\\
		\end{tabular}
	\end{ruledtabular}
	\fnt[1]{This work.}
	\fnt[2]{Unpublished data taken from \url{https://gw100.wordpress.com}.}
	\fnt[3]{Extrapolated values obtained from the def2-TZVP and def2-QZVP values.}
	\fnt[4]{Reference \onlinecite{Krause_2015}.}
	\fnt[5]{Experimental values taken from Ref.~\onlinecite{Maggio_2017}.}
\end{table*}


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
PFL would like to thank Fabien Bruneval for technical assistance. He also would like to thank Arjan Berger and Pina Romaniello for stimulating discussions.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%

\bibliography{GW-srDFT,GW-srDFT-control,biblio}

\end{document}
