314 lines
10 KiB
TeX
314 lines
10 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,hyperref,multirow,amsmath,amssymb,amsfonts,physics}
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\usepackage[normalem]{ulem}
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\usepackage[version=4]{mhchem}
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\usepackage[compat=1.1.0]{tikz-feynman}
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\usepackage[tracking]{microtype}
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\usepackage{mathpazo,libertine}
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\usepackage{algorithmicx,algorithm,algpseudocode}
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\algnewcommand\algorithmicassert{\texttt{assert}}
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\algnewcommand\Assert[1]{\State \algorithmicassert(#1)}
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\algrenewcommand{\algorithmiccomment}[1]{$\triangleright$ #1}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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%\usepackage{hyperref}
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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\newcommand{\eps}{\epsilon}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\mr}{\multirow}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
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% functionals, potentials, densities, etc
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\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
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\newcommand{\n}[1]{n^{#1}}
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\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
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\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
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% energies
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\Ecat}{E_\text{cat}}
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\newcommand{\Eneu}{E_\text{neu}}
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\newcommand{\Eani}{E_\text{ani}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\EFCI}{E_\text{FCI}}
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\newcommand{\br}{\mathbf{r}}
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\newcommand{\bw}{\mathbf{w}}
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\newcommand{\bP}[1]{\mathbf{P}^{#1}}
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\newcommand{\bG}{\mathbf{G}}
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\newcommand{\bH}{\mathbf{H}}
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\newcommand{\bF}[1]{\mathbf{F}^{#1}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
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%%% Manu %%%
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\definecolor{dgreen}{rgb}{0,.5,0}
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\newcommand{\manu}[1]{{\textcolor{dgreen}{ Manu: #1 }} }
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\newcommand{\be}{\begin{eqnarray}}
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\newcommand{\ee}{\end{eqnarray}}
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\begin{document}
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\title{Notes on density-functional theory for ensembles}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%
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\section{Optical gap}
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%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%
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\subsection{The weight-dependent functional}
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%%%%%%%%%%%%%%%%%%%%%
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The present weight-dependent density-functional approximation for ensemble (eDFA) is specifically designed for the calculation of excitation energies within eDFT.
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In order to take into bot single and double excitations simultaneously, the present eDFA is based on the ground state ($I=0$), first singly-excited state ($I=1$) and first doubly-excited states ($I=2$) of the two-electron ringium system.
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This functional can be seen as a local-density approximation (LDA) type functional for ensembles.
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All these states have the same (uniform) density $n = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
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One can easily write down the reduced (i.e.~per electron) Hartree+exchange (Hx) energy of these three states:
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\begin{align}
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\e{Hx}{(0)}(n) & = n,
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&
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\e{Hx}{(1)}(n) & = 4 n/3,
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&
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\e{Hx}{(2)}(n) & = 23 n/15.
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\end{align}
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Similarly, we can provide an accurate analytical expression of the correlation (c) energy via the following Pad\'e approximant
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\begin{equation}
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\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}}
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\end{equation}
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where $b^{(I)}$ and $c^{(I)}$ are state-dependent fitting parameters, which are provided in Table \ref{tab:OG_func}.
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$a^{(I)}$ is obtained via the high--density expansions of the correlation energy.
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%%% TABLE 1 %%%
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\begin{table*}
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\caption{\label{tab:OG_func}
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Parameters of the correlation functional for the optical gap.}
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\begin{ruledtabular}
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\begin{tabular}{lcddd}
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State & $I$ & \mcc{$a^{(I)}$} & \mcc{$b^{(I)}$} & \mcc{$c^{(I)}$} \\
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\hline
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Ground state & $0$ & -0.0137078 & 0.0748000 & 0.0566294 \\
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First singly-excited state & $1$ & -0.0240765 & 0.0553987 & 0.0163852 \\
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First doubly-excited state & $2$ & -0.00935749 & 0.0334572 & -0.0249377 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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Based on these quantities, one can build three-state weight-dependent functionals:
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\begin{subequations}
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\begin{align}
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\e{Hx}{\bw}(n) & = (1-w_1-w_2) \e{Hx}{(0)}(n) + w_1 \e{Hx}{(1)}(n) + w_2 \e{Hx}{(2)}(n),
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\\
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\e{c}{\bw}(n) & = (1-w_1-w_2) \e{c}{(0)}(n) + w_1 \e{c}{(1)}(n) + w_2 \e{c}{(2)}(n)
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\\
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\e{Hxc}{\bw}(n) & = \e{Hx}{\bw}(n) + \e{c}{\bw}(n),
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\end{align}
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\end{subequations}
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with $\bw = (w_1,w_2)$.
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%For notational convenience, we also define the corresponding potentials as
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%\begin{align}
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% \v{Hx}{\bw}(n) & = n\,\e{Hx}{\bw}(n)
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% &
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% \v{c}{\bw}(n) & = n\,\e{c}{\bw}(n)
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% &
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% \v{Hxc}{\bw}(n) & = n\,\e{Hxc}{\bw}(n)
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%\end{align}
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In the "c-only" case, the individual energies can be extracted as follows
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\begin{equation}
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\begin{split}
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E^{(I)} & = \frac{1}{2} \Tr[\bP{(I)} \cdot (\bH + \bF{(I)})]
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\\
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& + \int \qty{ \e{c}{\bw}[\n{\bw}(\br)] + \n{\bw}(\br) \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} } \n{(I)}(\br) d\br
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\\
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& + \LZ{c}{} + \DD{c}{(I)}.
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\end{split}
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\end{equation}
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where $\bH$ is the core Hamiltonian, $\bF{(I)}$ and $\bP{(I)}$ are the Fock and the density matrix associated with the $I$th state and $\n{(I)}(\br)$ its corresponding density.
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The correlation part of thr (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by
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\begin{align}
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\LZ{c}{} & = - \int \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
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\\
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\DD{c}{(I)} & = \sum_{J>0} (\delta_{IJ} - w_J) \int \left. \pdv{\e{c}{\bw}[\n{}]}{w_J}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br
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\end{align}
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where we note that, by construction,
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\begin{equation}
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\left. \pdv{\e{c}{\bw}[\n{}]}{w_J}\right|_{\n{} = \n{\bw}(\br)} = \e{c}{(J)}[\n{\bw}(\br)] - \e{c}{(0)}[\n{\bw}(\br)]
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\end{equation}
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In the case of a "Hxc" functional, we have
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\begin{equation}
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\begin{split}
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E^{(I)} & = \Tr[\bP{(I)} \cdot \bH]
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\\
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& + \int \qty{ \e{Hxc}{\bw}[\n{\bw}(\br)] + \n{\bw}(\br) \left. \fdv{\e{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} } \n{(I)}(\br) d\br
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\\
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& + \LZ{Hxc}{} + \DD{Hxc}{(I)}.
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\end{split}
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\end{equation}
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with
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\begin{align}
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\LZ{Hxc}{} & = - \int \left. \fdv{\e{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
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\\
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\DD{Hxc}{(I)} & = \sum_{J>0} (\delta_{IJ} - w_J) \int \left. \pdv{\e{Hxc}{\bw}[\n{}]}{w_J}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br
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\end{align}
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%%%%%%%%%%%%%%%%%%%%%
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\subsection{Implementation}
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%%%%%%%%%%%%%%%%%%%%%
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%%% FIGURE 1 %%%
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\begin{figure}
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\begin{algorithmic}[1]
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\Procedure{Optical Gap}{}
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\State
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\While{}
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\State
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\State
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% \For{}
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% \State
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% \EndFor
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\EndWhile
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% \If{}
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% \State
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% \EndIf
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\EndProcedure
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\end{algorithmic}
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\caption{
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\label{fig:algo_OG}
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Pseudo-code for the calculation of the optical gap within eDFT.
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$\tau$ is a user-defined threshold.
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}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%
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\section{Fundamental gap}
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%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%
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\subsection{The weight-dependent functional}
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%%%%%%%%%%%%%%%%%%%%%
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Similarly to what have been done for the optical gap, one can design easily designed a weight-dependent correlation eDFA for the fundamental gap.
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This time, our functional is based on the neutral ($I=0$), cation ($I=1$) and anion ($I=2$) species.
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These systems contain respectively, one, two and three electrons.
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Contrary to the optical gap, these three states have different densities which read
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\begin{align}
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\n{(0)} & = 2/(2\pi R),
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&
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\n{(1)} & = 1/(2\pi R),
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&
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\n{(2)} & = 3/(2\pi R).
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\end{align}
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Their reduced Hartree+exchange (Hx) energy are given by
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\begin{align}
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\e{Hx}{(0)}(n) & = n,
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&
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\e{Hx}{(1)}(n) & = 0,
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&
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\e{Hx}{(2)}(n) & = 40 n/27.
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\end{align}
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The expression of the correlation functional is identical to the previous case
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\begin{equation}
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\e{c}{(2)}(n) = \frac{a^{(2)}\,n}{n + b^{(2)} \sqrt{n} + c^{(2)}}
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\end{equation}
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where the coefficients are reported in Table \ref{tab:OG_func}.
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We note that $\e{c}{(0)}$ has been defined in the previous section and the one-electron cationic system has $\e{c}{(1)}(n) = 0$.
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%%% TABLE 2 %%%
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\begin{table*}
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\caption{\label{tab:OG_func}
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Parameters of the correlation functional for the optical gap.}
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\begin{ruledtabular}
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\begin{tabular}{lcddd}
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State & $I$ & \mcc{$a^{(I)}$} & \mcc{$b^{(I)}$} & \mcc{$c^{(I)}$} \\
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\hline
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Neutral & 0 & -0.0137078 & 0.0748000 & 0.0566294 \\
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Cation & 1 & 0 & 0 & 0 \\
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Anion & 2 & -0.0184842 & 0.0896216 & 0.0175732 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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Because of their difference in density, one needs to accommodate the weights in order to create a density-fixed eDFA.
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We have chosen the following form:
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\begin{equation}
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\e{c}{\bw}(n) = \qty(1 - \frac{N-1}{N} w_1 - \frac{N+1}{N} w_2) \e{c}{(0)} + w_1 \e{c}{(1)} + w_2 \e{c}{(2)}
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\end{equation}
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with, in our case, $N=2$.
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In the case of "c" functional, the indivdual energies are given by
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\begin{equation}
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\begin{split}
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E^{(I)} & = \frac{1}{2} \Tr[\bP{(I)} \cdot (\bH+\bF{(I)})]
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\\
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& + \int \qty{ \e{c}{\bw}[\n{\bw}(\br)] + \n{\bw}(\br) \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} } \n{(I)}(\br) d\br
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\\
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& + \LZ{c}{(I)} + \DD{c}{(I)}.
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\end{split}
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\end{equation}
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where we note that, now, the Levy-Zahariev shift is state dependent.
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\begin{align}
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\LZ{c}{(I)} & = - \int \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
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\\
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\DD{c}{(I)} & = \sum_{J>0} (\delta_{IJ} - w_J) \int \left. \pdv{\e{c}{\bw}[\n{}]}{w_J}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br
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\end{align}
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In the case of a "Hxc" functional for the fundamental gap, we have similar expressions as for the optical gap.
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%%% FIGURE 2 %%%
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\begin{figure}
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\begin{algorithmic}[1]
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\Procedure{Fundamental Gap}{}
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\State
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\While{}
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\State
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\State
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% \For{}
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% \State
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% \EndFor
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\EndWhile
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% \If{}
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% \State
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% \EndIf
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\EndProcedure
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\end{algorithmic}
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\caption{
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\label{fig:algo_FG}
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Pseudo-code for the calculation of the fundamental gap within eDFT.
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$\tau$ is a user-defined threshold.
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}
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\end{figure}
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\end{document}
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