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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}
\newcommand{\IUF}{Institut Universitaire de France, Paris, France}

\begin{document}	

\title{Supplementary Materials for ``A Density-Based Basis-Set Incompleteness Correction for GW Methods''}

\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Barth\'el\'emy 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}
\affiliation{\IUF}

\begin{abstract}
\end{abstract}

\maketitle

%Macros:
\newcommand{\basis}[0]{\mathcal{B}}
\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{srPBE}}^{\basis}[n]}
\newcommand{\epspbeueg}[0]{\bar{\varepsilon}^{\text{sr},\text{PBE}}_{\text{c,md}}}
\newcommand{\epspbe}[0]{\varepsilon^{\text{PBE}}_{\text{c}}}
\newcommand{\potpbeueg}[0]{\bar{v}_{\text{srPBE}}^{\basis}}
\newcommand{\potpbe}[0]{v^{\text{PBE}}_{\text{c}}}


\section{PBE-based complementary potential $\potpbeueg$}

Here, we provide the explicit expression of the PBE-based complementary potential in the case of closed-shell systems such as the ones studied in the present paper.
The PBE-based correlation energy functional with multideterminant reference (ECMD) has been previously reported in Ref.~\onlinecite{Loos_2019} and is defined by the following equation:
\begin{equation}
 \label{eq:def_pbe}
 \efuncbasispbe = \int n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) d\br{} ,
\end{equation}
with,
\begin{equation}
\label{eq:def_epsipbeueg}
  \epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3},
\end{equation}
where $\epspbe(n,s)$ is the usual PBE correlation functional \cite{Perdew_1996}, $s=\nabla n/n^{4/3}$ is the reduced density gradient,
\begin{equation}
 \beta(n,s) = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\epspbe(n,s)}{n_2^{\text{UEG}}(n)/n},
\end{equation}
and  
\begin{equation} 
 \label{eq:uegotop}
 n_2^{\text{UEG}}(n)=n^2g_0(r_s)
 \end{equation}
is the on-top pair density of the uniform electron gas (UEG). In Eq.~\eqref{eq:uegotop}, $r_s=(4\pi n/3)^{-1/3}$ the Wigner-Seitz radius and $g_0(r_s)$ is the UEG on-top pair-distribution function. The parametrization of $g_0(r_s)$ is given in Eq.~(46) of Ref.~\onlinecite{Gori-Giorgi_2006}.

The potential of this GGA ECMD complementary functional has the following form:
\begin{equation}
\begin{split}
  \potpbeueg[n]
  & = \fdv{\efuncbasispbe}{n} 
  \\
%  & = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
  & =\epspbeueg + n \pdv{\epspbeueg }{n}- \nabla \cdot \qty( n \pdv{\epspbeueg}{\nabla n} ).
  \end{split}
\end{equation}
Hence, we have to compute two main contributions: the scalar part $\pdv{\epspbeueg}{n}$ and the gradient part $\pdv{\epspbeueg }{\nabla n}$. 

\subsection{Scalar contribution}

For the scalar contribution, we simply differenciate Eq.~\eqref{eq:def_epsipbeueg} with respect to the density:
\begin{equation}
 \pdv{\epspbeueg }{n} 
 = \frac{\potpbe}{1+\beta\mu^3}
 - \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{n},
\end{equation}
where $\potpbe = \pdv{\epspbe}{n}$ and
\begin{equation}
 \pdv{\beta}{n} 
 = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}
 \Bigg[ \frac{\potpbe}{n_2^{\text{UEG}}/n}
 - \frac{\epspbe}{(n_2^{\text{UEG}}/n)^2} \frac{\partial (n_2^{\text{UEG}}/n)}{\partial n} \Bigg].
\end{equation}
The only remaining missing part is the derivative of $n_2^{\text{UEG}}/n$ with respect to the density:
\begin{equation}
\pdv{(n_2^{\text{UEG}}/n)}{n} = \pdv{[n g_0(r_s)]}{n} = g_0(r_s)+ n \pdv{g_0(r_s)}{n}.
\end{equation}
with
\begin{equation}
\pdv{g_0(r_s)}{n} = \pdv{r_s}{n} \pdv{g_0(r_s)}{r_s} = -(6 n^{2}\sqrt{\pi})^{-2/3} \pdv{g_0(r_s)}{r_s}.
\end{equation}
The derivative with respect to $r_s$ can be expressed as
\begin{equation}
\begin{aligned}
	\pdv{g_0(r_s)}{r_s} 
	& = \frac{e^{-F\,r_s}}{2} \big[ (-B + 2 C r_s + 3 D r_s^2 + 4 E r_s^3)  
	\\
	& - F (1 - B r_s + C r_s^2 + D r_s^3 + E r_s^4) \big],
\end{aligned}
\end{equation}
with 
 \begin{align}
  C &  = 0.0819306, \\
  F &  = 0.752411, \\
  D &  = -0.0127713,\\
  E &  =0.00185898,\\
  B &  = 0.7317 - F.
\end{align}


\subsection{Gradient contribution}

For the gradient part, we also used the chain rule:
\begin{equation}
 \pdv{\epspbeueg}{\nabla n} = \pdv{\epspbeueg}{\epspbe}\pdv{\epspbe}{\nabla n}.
\end{equation}
The term $\pdv{\epspbe}{\nabla n}$ is already known (\textbf{ref??}), and the partial derivative of $\epspbeueg$ with respect to $\epspbe$ is
\begin{equation}
 \pdv{\epspbeueg}{\epspbe} 
 = \frac{1}{1+\beta\mu^3}
 - \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{\epspbe},
\end{equation}
where 
\begin{equation}
 \pdv{\beta}{\epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
\end{equation}

\begin{figure*}
	\includegraphics[width=\linewidth]{IP_G0W0HF}
	\caption{
	IPs (in eV) computed at the {\GOWO}@HF (black circles), {\GOWO}@HF+srLDA (red squares), and {\GOWO}@HF+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set. 
	The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
	\label{fig:IP_G0W0HF}
	}
\end{figure*}

\begin{figure*}
	\includegraphics[width=\linewidth]{IP_G0W0PBE0}
	\caption{
	IPs (in eV) computed at the {\GOWO}@PBE0 (black circles), {\GOWO}@PBE0+srLDA (red squares), and {\GOWO}@PBE0+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set. 
	The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
	\label{fig:IP_G0W0HF}
	}
\end{figure*}


\bibliography{../GW-srDFT,../GW-srDFT-control}

\end{document}
