clean up barth pot
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\maketitle
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%Macros:
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\newcommand{\basis}[0]{\mathcal{B}}
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\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{srPBE}}^{\basis}[n]}
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\newcommand{\epspbeueg}[0]{\bar{\varepsilon}^{\text{sr},\text{PBE}}_{\text{c,md}}}
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\newcommand{\epspbe}[0]{\varepsilon^{\text{PBE}}_{\text{c}}}
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\newcommand{\potpbeueg}[0]{\bar{v}_{\text{srPBE}}^{\basis}}
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\newcommand{\potpbe}[0]{v^{\text{PBE}}_{\text{c}}}
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\section{PBE-based complementary potential $\potpbeueg$}
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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.
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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:
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\begin{equation}
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\label{eq:def_pbe}
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\efuncbasispbe = \int n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) d\br{} ,
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\end{equation}
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with,
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\begin{equation}
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\label{eq:def_epsipbeueg}
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\epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3},
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\end{equation}
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where $\epspbe(n,s)$ is the usual PBE correlation functional \cite{Perdew_1996}, $s=\nabla n/n^{4/3}$ is the reduced density gradient,
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\begin{equation}
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\beta(n,s) = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\epspbe(n,s)}{n_2^{\text{UEG}}(n)/n},
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\end{equation}
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and
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\begin{equation}
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\label{eq:uegotop}
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n_2^{\text{UEG}}(n)=n^2g_0(r_s)
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\end{equation}
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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}.
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The potential of this GGA ECMD complementary functional has the following form:
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\begin{equation}
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\begin{split}
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\potpbeueg[n]
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& = \fdv{\efuncbasispbe}{n}
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\\
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% & = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
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& =\epspbeueg + n \pdv{\epspbeueg }{n}- \nabla \cdot \qty( n \pdv{\epspbeueg}{\nabla n} ).
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\end{split}
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\end{equation}
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Hence, we have to compute two main contributions: the scalar part $\pdv{\epspbeueg}{n}$ and the gradient part $\pdv{\epspbeueg }{\nabla n}$.
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\subsection{Scalar contribution}
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For the scalar contribution, we simply differenciate Eq.~\eqref{eq:def_epsipbeueg} with respect to the density:
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\begin{equation}
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\pdv{\epspbeueg }{n}
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= \frac{\potpbe}{1+\beta\mu^3}
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- \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{n},
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\end{equation}
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where $\potpbe = \pdv{\epspbe}{n}$ and
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\begin{equation}
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\pdv{\beta}{n}
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= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}
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\Bigg[ \frac{\potpbe}{n_2^{\text{UEG}}/n}
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- \frac{\epspbe}{(n_2^{\text{UEG}}/n)^2} \frac{\partial (n_2^{\text{UEG}}/n)}{\partial n} \Bigg].
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\end{equation}
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The only remaining missing part is the derivative of $n_2^{\text{UEG}}/n$ with respect to the density:
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\begin{equation}
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\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}.
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\end{equation}
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with
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\begin{equation}
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\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}.
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\end{equation}
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The derivative with respect to $r_s$ can be expressed as
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\begin{equation}
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\begin{aligned}
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\pdv{g_0(r_s)}{r_s}
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& = \frac{e^{-F\,r_s}}{2} \big[ (-B + 2 C r_s + 3 D r_s^2 + 4 E r_s^3)
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\\
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& - F (1 - B r_s + C r_s^2 + D r_s^3 + E r_s^4) \big],
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\end{aligned}
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\end{equation}
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with
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\begin{align}
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C & = 0.0819306, \\
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F & = 0.752411, \\
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D & = -0.0127713,\\
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E & =0.00185898,\\
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B & = 0.7317 - F.
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\end{align}
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\subsection{Gradient contribution}
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For the gradient part, we also used the chain rule:
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\begin{equation}
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\pdv{\epspbeueg}{\nabla n} = \pdv{\epspbeueg}{\epspbe}\pdv{\epspbe}{\nabla n}.
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\end{equation}
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The term $\pdv{\epspbe}{\nabla n}$ is already known (\textbf{ref??}), and the partial derivative of $\epspbeueg$ with respect to $\epspbe$ is
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\begin{equation}
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\pdv{\epspbeueg}{\epspbe}
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= \frac{1}{1+\beta\mu^3}
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- \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{\epspbe},
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\end{equation}
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where
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\begin{equation}
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\pdv{\beta}{\epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
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\end{equation}
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\begin{figure*}
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\includegraphics[width=\linewidth]{IP_G0W0HF}
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\caption{
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@ -1,187 +0,0 @@
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%% ****** Start of file auguide.tex ****** %
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%%
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%% This file is part of the AIP distribution of substyles for REVTeX 4.1
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%% For version 4.1r of REVTeX, August 2010
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%%
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%% Copyright (c) 2009,2010 American Institute of Physics
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%%
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%\listfiles
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\documentclass[
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reprint,
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amssymb, amsmath,
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aip,jcp
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]{revtex4-1}
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\usepackage{comment}
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\usepackage{dcolumn}
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%\usepackage{docs}%
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%\usepackage{bm}%
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\usepackage[colorlinks=true,linkcolor=blue]{hyperref}%
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\expandafter\ifx\csname package@font\endcsname\relax\else
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\expandafter\expandafter
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\expandafter\usepackage
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\expandafter\expandafter
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\expandafter{\csname package@font\endcsname}%
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\fi
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\hyphenation{title}
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\usepackage{xspace}
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\usepackage{graphicx}
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%\usepackage{subfig}
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\usepackage[version=3]{mhchem}
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\parskip=0.1in
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\usepackage{amsmath}
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\usepackage[normalem]{ulem}
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\usepackage[utf8]{inputenc}
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%Macros:
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\newcommand{\basis}[0]{\mathcal{B}}
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\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{srPBE}}^{\basis}[n]}
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\newcommand{\epspbeueg}[0]{\bar{\varepsilon}^{\text{sr},\text{PBE}}_{\text{c,md}}}
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\newcommand{\epspbe}[0]{\varepsilon^{\text{PBE}}_{\text{c}}}
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\newcommand{\potpbeueg}[0]{\bar{v}_{\text{srPBE}}^{\basis}}
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\newcommand{\potpbe}[0]{v^{\text{PBE}}_{\text{c}}}
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\setcounter{secnumdepth}{4}
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\begin{document}
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\section{PBE-based complementary potential $\potpbeueg$}
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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:
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\begin{equation}
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\label{eq:def_pbe}
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\efuncbasispbe = \int \, \text{d}{\bf r} \,\,n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}({\bf r})),
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\end{equation}
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with,
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\begin{equation}
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\label{eq:def_epsipbeueg}
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\epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3},
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\end{equation}
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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,
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\begin{equation}
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\beta(n,s) = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\epspbe(n,s)}{n_2^{\text{UEG}}(n)/n},
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\end{equation}
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and where
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\begin{equation}
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\label{eq:uegotop}
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n_2^{\text{UEG}}(n)=n^2(1-\xi^2)g_0(r_s),
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\end{equation}
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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}.
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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:
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\begin{equation}
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\label{eq:uegotop2}
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n_2^{\text{UEG}}(n)=n^2 g_0(r_s).
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\end{equation}
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The potential of this GGA-based functional has the following form:
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\begin{equation}
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\begin{aligned}
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& \potpbeueg[n]({\bf r},\mu) = \frac{\delta \efuncbasispbe}{\delta n({\bf r})} \\
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& = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
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& =\epspbeueg +n\frac{\partial \epspbeueg }{\partial n}- \nabla . n\frac{\partial \epspbeueg }{\partial \nabla n}.
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\end{aligned}
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\end{equation}
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So we have to compute two main contributions, the scalar part $\frac{\partial \epspbeueg }{\partial n}$ and the gradient part $\frac{\partial \epspbeueg }{\partial \nabla n}$.
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$\bullet$ For the scalar contribution, we simply derived eq.~\ref{eq:def_epsipbeueg} with respect to the density:
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\begin{equation}
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\frac{\partial \epspbeueg }{\partial n}=\frac{\potpbe(1+\beta \mu^3)-\epspbe \frac{\partial\beta}{\partial n}\mu^3}{(1+\beta\mu^3)^2},
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\end{equation}
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Where
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\begin{equation}
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\potpbe[n]({\bf r}) =\frac{\partial\epspbe}{\partial n}
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\end{equation}
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and
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\begin{equation}
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\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}.
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\end{equation}
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The only remaining part is the derivative of $n_2^{\text{UEG}}/n$ with respect to the density:
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\begin{equation}
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\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}.
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\end{equation}
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To compute $\frac{\partial g_0(r_s)}{\partial n}$, we used the chain rule:
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\begin{equation}
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\frac{\partial g_0(r_s)}{\partial n} = \frac{\partial g_0(r_s)}{\partial r_s}\frac{\partial r_s}{\partial n}.
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\end{equation}
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The derivative with respect to $r_s$ can be express:
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\begin{equation}
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\begin{aligned}
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&\frac{\partial g_0(r_s)}{\partial r_s} = \\
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& 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) \\
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& -(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))),
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\end{aligned}
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\end{equation}
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with
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\begin{equation}
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\begin{aligned}
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& C_{g_0} = 0.0819306, \\
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& F_{g_0} = 0.752411, \\
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& D_{g_0} = -0.0127713,\\
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& E_{g_0} =0.00185898,\\
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& B_{g_0} = 0.7317 - F_{g_0}.
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\end{aligned}
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\end{equation}
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And finally the derivative of $r_s$ with respect to $n$ is equal to:
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\begin{equation}
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\frac{\partial g_0(r_s)}{\partial n} = -(6^{2/3}n^{4/3}\pi^{1/3})^{-1}.
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\end{equation}
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$\bullet$ For the gradient part, we also used the chain rule:
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\begin{equation}
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\frac{\partial \epspbeueg}{\partial \nabla n}=\frac{\partial \epspbeueg}{\partial \epspbe}\frac{\partial \epspbe}{\partial \nabla n}.
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\end{equation}
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$\frac{\partial \epspbe}{\partial \nabla n}$ is already known (\textbf{Quelqu'un a une ref pour ça??}), and the partial derivative of $\epspbeueg$ with respect to $\epspbe$ is trivial:
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\begin{equation}
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\frac{\partial \epspbeueg}{\partial \epspbe}= \frac{(1+\beta \mu^3)-\epspbe \frac{\partial\beta}{\partial \epspbe}\mu^3}{(1+\beta\mu^3)^2}
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\end{equation}
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where
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\begin{equation}
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\frac{\partial \beta}{\partial \epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
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\end{equation}
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\bibliography{paper}
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\end{document}
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