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\newcommand{\HF}{\text{HF}}
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\newcommand{\evGW}{ev$GW$}	
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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{Complementary short-range correlation potentials}

Here, we provide the expressions of the complementary short-range LDA and PBE correlation potentials used in the present work in the case of closed-shell systems.

\subsection{Complementary short-range LDA correlation potential}

The complementary short-range LDA correlation energy functional with multideterminant reference has the expression~\cite{Toulouse_2005,Paziani_2006}
\begin{equation}
        \label{eq:def_lda_tot}
        \bE{\srLDA}{\Bas}[\n{}{}] =
        \int \n{}{}(\br{}) \be{\text{c,md}}{\srLDA}(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation}
with
\begin{equation}
  \be{\text{c,md}}{\srLDA}(\n{}{},\rsmu{}{}) = \be{\text{c}}{\srLDA}(\n{}{},\rsmu{}{}) + \Delta^{\text{lr-sr}}(n,\mu),
\end{equation}
with $\be{\text{c,md}}{\srLDA}(\n{}{},\rsmu{}{})$ is the complementary short-range LDA correlation energy functional (with single-determinant reference) and $\Delta^{\text{lr-sr}}(n,\mu)$ is a mixed long-range/short-range contribution, both parametrized in Ref.~\onlinecite{Paziani_2006}.

The corresponding complementary srLDA potential is
\begin{eqnarray}
\bpot{\srLDA}{\Bas}[\n{}{}](\br{}) &=& \frac{\delta \bE{\srLDA}{\Bas}[\n{}{}]}{\delta \n{}{}(\br{})}
\nonumber\\
 &=& \be{\text{c,md}}{\srLDA}(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) 
\nonumber\\
&&+ n(\br{}) \frac{\partial \be{\text{c,md}}{\srLDA}}{\partial n} (\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})).
\end{eqnarray}
The density derivative of $\be{\text{c,md}}{\srLDA}$ is calculated as
\begin{eqnarray}
\frac{\partial \be{\text{c,md}}{\srLDA}}{\partial n} = \frac{\partial \be{\text{c}}{\srLDA}}{\partial n} + \frac{\partial \Delta^{\text{lr-sr}}}{\partial n},
\end{eqnarray}
where $\partial \be{\text{c}}{\srLDA}/\partial n$ is given as a subroutine on Paola Gori-Giorgi's web site (\url{https://www.quantummatter.eu/source-codes-2}) and we have calculated $\partial \Delta^{\text{lr-sr}}/\partial n$ by taking the derivative of Eq. (42) of Ref.~\onlinecite{Paziani_2006}.

\subsection{Complementary short-range PBE correlation potential}

The complementary short-range PBE correlation energy functional with multideterminant reference has the expression~\cite{Loos_2019}
\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}
Here, $\epspbe(n,s)$ is the usual PBE correlation functional \cite{Perdew_1996}, $s$ 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_\text{s})
 \end{equation}
is the on-top pair density of the uniform electron gas (UEG). In Eq.~\eqref{eq:uegotop}, $g_0(r_\text{s})$ is the UEG on-top pair-distribution function written as a function of the Wigner-Seitz radius $r_\text{s}=(4\pi n/3)^{-1/3}$. We use the parametrization of $g_0(r_\text{s})$ given in Eq.~(46) of Ref.~\onlinecite{Gori-Giorgi_2006}.

The corresponding complementary srPBE potential is
\begin{eqnarray}
  \potpbeueg[n](\br{})
   &=& \fdv{\efuncbasispbe}{n(\br{})} 
\nonumber\\
  &=& \epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) 
\nonumber\\
  &+& n(\br{}) \pdv{\epspbeueg }{n} (n({\bf r}),s({\bf r}),\mu^{\basis}(\br{}))
\nonumber\\
&-& \nabla \cdot \qty( n(\br{}) \pdv{\epspbeueg}{\nabla n} (n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) ).\,\,\,
\end{eqnarray}
Hence, we have to compute the density derivative $\partial \epspbeueg/\partial n$ and the density-gradient derivative $\partial \epspbeueg/\partial \nabla n$.

\subsubsection{Density derivative}

From Eq.~\eqref{eq:def_epsipbeueg}, the density derivative is found to be
\begin{equation}
 \pdv{\epspbeueg }{n} 
 = \frac{1}{1+\beta\mu^3} \pdv{\epspbe}{n}
 - \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{n},
\end{equation}
where $\partial \epspbe/\partial n$ is the density derivative of the usual PBE correlation functional, and 
\begin{eqnarray}
 \pdv{\beta}{n} 
 &=& \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}
 \Bigg[ \frac{1}{n_2^{\text{UEG}}/n} \pdv{\epspbe}{n}
\nonumber\\
 &&\phantom{xxxxx} - \frac{\epspbe}{(n_2^{\text{UEG}}/n)^2} \frac{\partial (n_2^{\text{UEG}}/n)}{\partial n} \Bigg].
\end{eqnarray}
The only remaining missing part is the derivative of $n_2^{\text{UEG}}/n$ which is
\begin{equation}
\pdv{(n_2^{\text{UEG}}/n)}{n} = \pdv{[n g_0(r_\text{s})]}{n} = g_0(r_\text{s})+ n \pdv{g_0(r_\text{s})}{n},
\end{equation}
with
\begin{equation}
\pdv{g_0(r_\text{s})}{n} = \pdv{r_\text{s}}{n} \pdv{g_0(r_\text{s})}{r_\text{s}} = -(6 n^{2}\sqrt{\pi})^{-2/3} \pdv{g_0(r_\text{s})}{r_\text{s}}.
\end{equation}
Finally, we calculate $\partial g_0(r_\text{s}) /\partial r_\text{s}$ by taking the derivative of Eq.~(46) of Ref.~\onlinecite{Gori-Giorgi_2006}
\begin{equation}
\begin{aligned}
	\pdv{g_0(r_\text{s})}{r_\text{s}} 
	& = \frac{e^{-F\,r_\text{s}}}{2} \big[ (-B + 2 C r_\text{s} + 3 D r_\text{s}^2 + 4 E r_\text{s}^3)  
	\\
	& - F (1 - B r_\text{s} + C r_\text{s}^2 + D r_\text{s}^3 + E r_\text{s}^4) \big],
\end{aligned}
\end{equation}
with $C   = 0.0819306$, $F   = 0.752411$, $D   = -0.0127713$, $E   =0.00185898$, and $B   = 0.7317 - F$.

\subsubsection{Density-gradient derivative}

For the density-gradient derivative, we use the chain rule
\begin{equation}
 \pdv{\epspbeueg}{\nabla n} = \pdv{\epspbeueg}{\epspbe}\pdv{\epspbe}{\nabla n},
\end{equation}
where $\partial \epspbe/\partial \nabla n$ is the density-gradient derivative of the usual PBE correlation functional, and
\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}
with
\begin{equation}
 \pdv{\beta}{\epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
\end{equation}

\section{Additional graphs of the convergence of the IPs of the GW20 subset}

Graphs reporting the convergence of the IPs of each molecule of the GW20 subset at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels are given in Figure~\ref{fig:IP_G0W0HF} and~\ref{fig:IP_G0W0PBE0}, respectively.

\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_G0W0PBE0}
	}
\end{figure*}


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

\end{document}