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b/FarDFT/Manuscript/FarDFT.tex index be6aecb..4aee5e6 100644 --- a/FarDFT/Manuscript/FarDFT.tex +++ b/FarDFT/Manuscript/FarDFT.tex @@ -1,5 +1,6 @@ \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem} +\usepackage{libertine} \usepackage[ colorlinks=true, @@ -50,7 +51,6 @@ \newcommand{\n}[2]{n_{#1}^{#2}} \newcommand{\DD}[2]{\Delta_\text{#1}^{#2}} \newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}} -\newcommand{\Cx}[1]{C_\text{x}^{#1}} % energies \newcommand{\EHF}{E_\text{HF}} @@ -60,7 +60,6 @@ \newcommand{\Eani}{E_\text{ani}} \newcommand{\EPT}{E_\text{PT2}} \newcommand{\EFCI}{E_\text{FCI}} -\newcommand{\LDA}{\text{LDA}} % matrices \newcommand{\br}{\bm{r}} @@ -97,6 +96,7 @@ \newcommand{\kcal}{kcal/mol} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France} \begin{document} @@ -104,8 +104,11 @@ \author{Clotilde \surname{Marut}} \affiliation{\LCPQ} +\author{Emmanuel Fromager} +\email{fromagere@unistra.fr} +\affiliation{\LCQ} \author{Pierre-Fran\c{c}ois \surname{Loos}} -\email{loos@irsamc.ups-tlse.fr} +\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \begin{abstract} @@ -163,7 +166,7 @@ Here, we restrict our study to spin-unpolarized systems, \ie, $\n{\uparrow}{} = The present weight-dependent eDFA is specifically designed for the calculation of double excitations within eDFT. As mentioned previously, we consider a two-state ensemble including the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the two-electron glomium system. -All these states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome where the electrons are confined. +All these states have the same (uniform) density $\n{}{} = 2/(2\pi/2 R^3)$ where $R$ is the radius of the glome where the electrons are confined. We refer the interested reader to Refs.~\onlinecite{Loos_2011b} for more details about this paradigm. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -178,66 +181,27 @@ The reduced (\ie, per electron) HF energy for these two states is \e{HF}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}. \end{align} \end{subequations} -These two energies can be conveniently decomposed as -\begin{equation} - \e{HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{H}{(0)}(\n{}{}) + \e{x}{(I)}(\n{}{}), -\end{equation} -with +These two energies can be conveniently decomposed as \begin{subequations} \begin{align} - \kin{s}{(0)}(\n{}{}) & = 0, - & - \kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3}. + \kin{s}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3}, \\ - \e{H}{(0)}(\n{}{}) & = \frac{8}{3\pi^{1/3}} \n{}{1/3}, - & - \e{H}{(1)}(\n{}{}) & = \frac{352}{105\pi^{1/3}} \n{}{1/3}. - \\ - \e{x}{(0)}(\n{}{}) & = - \frac{4}{3\pi^{1/3}} \n{}{1/3}, - & - \e{x}{(1)}(\n{}{}) & = - \frac{176}{105\pi^{1/3}} \n{}{1/3}. + \kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}. \end{align} \end{subequations} -Knowing that the exchange functional has the following form -\begin{equation} - \e{x}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3} -\end{equation} -we obtain -\begin{align} - \Cx{(0)} & = - \frac{4}{3} \qty( \frac{2}{\pi} )^{1/3}, - & - \Cx{(1)} & = - \frac{176}{105} \qty( \frac{2}{\pi} )^{1/3} -\end{align} -We can now combine these two exchange functionals to create a weight-dependent exchange functional -\begin{equation} -\begin{split} - \e{x}{\ew{}}(\n{}{}) - & = (1-\ew{}) \e{x}{(0)}(\n{}{}) + \ew{} \e{x}{(1)}(\n{}{}) - \\ - & = \Cx{\ew{}} \n{}{1/3} -\end{split} -\end{equation} -with -\begin{equation} - \Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)} -\end{equation} -Amazingly, the weight dependence of the exchange functional can be transfered to the Subscript[C, x] coefficient. -This is obvious but kind of nice. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Weight-dependent correlation functional} \label{sec:Ec} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Based on highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant +Based on highly-accurate calculations (see below), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \begin{equation} \label{eq:ec} - \e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}}, + \e{xc}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}}, \end{equation} -where the $a_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}. -The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a} +where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}. +The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a} Equation \eqref{eq:ec} provides two state-specific correlation DFAs based on a two-electron system. Combining these, one can build a two-state weight-dependent correlation eDFA: \begin{equation} @@ -249,7 +213,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA: \begin{figure} % \includegraphics[width=\linewidth]{Ec} \caption{ - Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = ...$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system. + Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi n)$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system. The data gathered in Table \ref{tab:Ref} are also reported. } \label{fig:Ec} @@ -260,7 +224,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA: \begin{table} \caption{ \label{tab:Ref} - $-\e{c}{(I)}$ as a function of the radius of the glome $R$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system. + $-\e{c}{(I)}$ as a function of the radius of the ring $R$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system. } \begin{ruledtabular} \begin{tabular}{ldd} @@ -279,6 +243,8 @@ Combining these, one can build a two-state weight-dependent correlation eDFA: $20$ & & \\ $50$ & & \\ $100$ & & \\ + $150$ & & \\ + $200$ & & \\ \end{tabular} \end{ruledtabular} \end{table} @@ -286,7 +252,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA: Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \begin{equation} \label{eq:ec} - \e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}}, + \e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}}{1 + c_2^{(I)} \n{}{-1/6} + c_3^{(I)} \n{}{-1/3}}, \end{equation} where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript. The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a} @@ -301,10 +267,10 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.} \begin{ruledtabular} \begin{tabular}{lcddd} - State & $I$ & \tabc{$a_1^{(I)}$} & \tabc{$a_2^{(I)}$} & \tabc{$a_3^{(I)}$} \\ + State & $I$ & \tabc{$c_1^{(I)}$} & \tabc{$c_2^{(I)}$} & \tabc{$c_3^{(I)}$} \\ \hline - Ground state & $0$ & -0.0238184 & +0.00575719 & +0.0830576 \\ - Doubly-excited state & $1$ & -0.0144633 & -0.0504501 & +0.0331287 \\ + Ground state & $0$ & & & \\ + Doubly-excited state & $1$ & & & \\ \end{tabular} \end{ruledtabular} \end{table*} @@ -320,30 +286,21 @@ In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more \end{equation} where \begin{equation} - \be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\LDA}(\n{}{}) - \e{xc}{(0)}(\n{}{}). + \be{xc}{(I)}(\n{}{}) = \e{xc}{(I)}(\n{}{}) + \e{xc}{\text{LDA}}(\n{}{}) - \e{xc}{(0)}(\n{}{}). \end{equation} The local-density approximation (LDA) exchange-correlation functional is \begin{equation} - \e{xc}{\LDA}(\n{}{}) = \e{x}{\LDA}(\n{}{}) + \e{c}{\LDA}(\n{}{}). + \e{xc}{\text{LDA}}(\n{}{}) = \e{x}{\text{LDA}}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}). \end{equation} -where we use here the Dirac exchange functional and the VWN5 correlation functional -\begin{subequations} -\begin{align} - \e{x}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3} - \\ - \e{c}{\LDA}(\n{}{}) & \equiv \e{c}{\text{VWN5}}(\n{}{}). -\end{align} -\end{subequations} -with $\Cx{\LDA} = -\frac{3}{2} \qty(\frac{3}{4\pi})^{1/3}$. - + Equation \eqref{eq:becw} can be recast \begin{equation} \label{eq:eLDA} \be{xc}{\ew{}}(\n{}{}) - = \e{xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})], + = \e{xc}{\text{LDA}}(\n{}{}) + \ew{} \qty[\e{xc}{(1)}(\n{}{})-\e{xc}{(0)}(\n{}{})], \end{equation} which nicely highlights the centrality of the LDA in the present eDFA. -In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\LDA}(\n{}{})$. +In particular, $\be{xc}{(0)}(\n{}{}) = \e{xc}{\text{LDA}}(\n{}{})$. Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles. This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014}