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\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}}
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\begin{document}



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


The PBE-based multideterminant short-range correlation complementary density functional used in this paper is the one presented in Ref.~\onlinecite{LooPraSce} and which is defined with the following equation:

\begin{equation}
 \label{eq:def_pbe}
 \efuncbasispbe = \int \, \text{d}{\bf r} \,\,n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}({\bf r})),
\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 $\epspbe$ is the usual PBE correlation functional~\cite{pbe}, $s({\bf r})=\nabla n({\bf r})/n({\bf r})^{4/3}$ 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 where 
 
\begin{equation} 
 \label{eq:uegotop}
 n_2^{\text{UEG}}(n)=n^2(1-\xi^2)g_0(r_s),
 \end{equation}
 
is the on-top pair density of the uniform electron gas. In eq.~\ref{eq:uegotop}, $\xi=(n_{\uparrow}-n_{\downarrow})/n$ is the spin polarisation,$r_s=(\frac{4\pi n}{3})^{-1/3}$ the Wigner-Seitz radius and $g_0(r_s)$ is the UEG on-top pair-distribution function. The parametrization used in this paper for this last function is detailed in eq.46 of Ref.~\onlinecite{GorSav2006}.

In the present investigation, we have only studied close shell cases for which $n_{\uparrow} = n_{\downarrow}$, which implies that $\xi = 0$. The on-top pair density of the uniform electron gas can thus be rewritten:

\begin{equation} 
 \label{eq:uegotop2}
 n_2^{\text{UEG}}(n)=n^2 g_0(r_s).
 \end{equation}

The potential of this GGA-based functional has the following form:

\begin{equation}
 \begin{aligned}
  & \potpbeueg[n]({\bf r},\mu)  = \frac{\delta \efuncbasispbe}{\delta n({\bf r})} \\
  & = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
  & =\epspbeueg +n\frac{\partial \epspbeueg }{\partial n}- \nabla . n\frac{\partial  \epspbeueg }{\partial \nabla n}.
  \end{aligned}
\end{equation}

So we have to compute two main contributions, the scalar part $\frac{\partial \epspbeueg }{\partial n}$ and the gradient part $\frac{\partial  \epspbeueg }{\partial \nabla n}$. 


$\bullet$ For the scalar contribution, we simply derived eq.~\ref{eq:def_epsipbeueg} with respect to the density:

\begin{equation}
 \frac{\partial \epspbeueg }{\partial n}=\frac{\potpbe(1+\beta \mu^3)-\epspbe  \frac{\partial\beta}{\partial n}\mu^3}{(1+\beta\mu^3)^2},
\end{equation}

Where 

\begin{equation}
 \potpbe[n]({\bf r}) =\frac{\partial\epspbe}{\partial n} 
\end{equation}

and

\begin{equation}
 \frac{\partial \beta}{\partial n}=\frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\potpbe (n_2^{\text{UEG}}/n)-\epspbe \frac{\partial n_2^{\text{UEG}}/n}{\partial n}}{(n_2^{\text{UEG}}/n)^2}.
\end{equation}

The only remaining part is the derivative of $n_2^{\text{UEG}}/n$ with respect to the density:

\begin{equation}
\frac{\partial n_2^{\text{UEG}}/n}{\partial n} = \frac{\partial n g_0(r_s)}{\partial n} = g_0(r_s)+ n\frac{\partial g_0(r_s)}{\partial n}.
\end{equation}


To compute $\frac{\partial g_0(r_s)}{\partial n}$, we used the chain rule:

\begin{equation}
\frac{\partial g_0(r_s)}{\partial n} = \frac{\partial g_0(r_s)}{\partial r_s}\frac{\partial r_s}{\partial n}.
\end{equation}

The derivative with respect to $r_s$ can be express:

\begin{equation}
\begin{aligned}
&\frac{\partial g_0(r_s)}{\partial r_s} = \\
& 0.5e^{-F_{g_0}*r_s} ( (-B_{g_0}+2C_{g_0}r_s+3D_{g_0}*r_s^2+4E_{g_0}r_s^3)  \\
& -(F_{g_0}(1 - B_{g_0}r_s + C_{g_0}r_s^2 + D_{g_0}rs^3 + E_{g_0}r_s^4))),
\end{aligned}
\end{equation}
 
 with
 
 \begin{equation}
 \begin{aligned}
  &  C_{g_0} = 0.0819306, \\
  &  F_{g_0} = 0.752411, \\
  &  D_{g_0} = -0.0127713,\\
  &  E_{g_0} =0.00185898,\\
  &  B_{g_0} = 0.7317 - F_{g_0}.
  \end{aligned}
\end{equation}

And finally the derivative of $r_s$ with respect to $n$ is equal to:

\begin{equation}
\frac{\partial g_0(r_s)}{\partial n} =  -(6^{2/3}n^{4/3}\pi^{1/3})^{-1}.
\end{equation}

$\bullet$ For the gradient part, we also used the chain rule:

\begin{equation}
 \frac{\partial \epspbeueg}{\partial \nabla n}=\frac{\partial \epspbeueg}{\partial \epspbe}\frac{\partial \epspbe}{\partial \nabla n}.
\end{equation}

$\frac{\partial \epspbe}{\partial \nabla n}$ is already known (\textbf{Quelqu'un a une ref pour ça??}), and the partial derivative of $\epspbeueg$ with respect to $\epspbe$ is trivial:

\begin{equation}
 \frac{\partial \epspbeueg}{\partial \epspbe}= \frac{(1+\beta \mu^3)-\epspbe  \frac{\partial\beta}{\partial \epspbe}\mu^3}{(1+\beta\mu^3)^2}
\end{equation}

where 

\begin{equation}
 \frac{\partial \beta}{\partial \epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
\end{equation}


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