clean up barth pot

JCTC_revision/SI/GW-srDFT-SI.tex

@@ -181,6 +181,110 @@
 
 \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{

JCTC_revision/SI/paper.tex

@@ -1,187 +0,0 @@
-
-%% ****** Start of file auguide.tex ****** %
-%%
-%%   This file is part of the AIP distribution of substyles for REVTeX 4.1
-%%   For version 4.1r of REVTeX, August 2010
-%%
-%%   Copyright (c) 2009,2010 American Institute of Physics
-%%
-%\listfiles
-
-\documentclass[
- reprint,
- amssymb, amsmath,
- aip,jcp
-]{revtex4-1}
-
-\usepackage{comment} 
-\usepackage{dcolumn} 
-%\usepackage{docs}%
-%\usepackage{bm}%
-\usepackage[colorlinks=true,linkcolor=blue]{hyperref}%
-\expandafter\ifx\csname package@font\endcsname\relax\else
- \expandafter\expandafter
- \expandafter\usepackage
- \expandafter\expandafter
- \expandafter{\csname package@font\endcsname}%
-\fi
-\hyphenation{title}
-\usepackage{xspace}
-
-\usepackage{graphicx}
-%\usepackage{subfig}
-
-\usepackage[version=3]{mhchem}
-\parskip=0.1in
- 
-\usepackage{amsmath}
-
-\usepackage[normalem]{ulem}
-\usepackage[utf8]{inputenc}
-
-%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}}}
-
-\setcounter{secnumdepth}{4}
-\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}
-
-
-\bibliography{paper}
-
- \end{document}