diff --git a/JCTC_revision/SI/GW-srDFT-SI.tex b/JCTC_revision/SI/GW-srDFT-SI.tex index e69288e..ed55058 100644 --- a/JCTC_revision/SI/GW-srDFT-SI.tex +++ b/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{ diff --git a/JCTC_revision/SI/paper.tex b/JCTC_revision/SI/paper.tex deleted file mode 100644 index 2b17f97..0000000 --- a/JCTC_revision/SI/paper.tex +++ /dev/null @@ -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}