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\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}}
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\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}}
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\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}}}
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\newcommand{\bde}{\boldsymbol{\Delta\epsilon}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\fc}{f_\text{c}}
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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{Bath\'elemy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
\author{Emmanuel Giner}
\affiliation{\LCT}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Julien Toulouse}
\affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}

\begin{abstract}
%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
%	\centering
%  \includegraphics[width=\linewidth]{TOC}
%\end{wrapfigure}
\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}
%%%%%%%%%%%%%%%%%%%%%%%%



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

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

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), CALMIP (Toulouse) under allocation 2019-18005 and the Jarvis-Alpha cluster from the \textit{Institut Parisien de Chimie Physique et Th\'eorique}.
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%

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

\end{document}