929 lines
40 KiB
TeX
929 lines
40 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
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\usepackage{mathpazo,libertine}
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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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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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%useful stuff
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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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{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\la}{\lambda}
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\newcommand{\si}{\sigma}
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% functionals, potentials, densities, etc
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\newcommand{\eps}{\epsilon}
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\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
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\renewcommand{\v}[2]{v_\text{#1}^{#2}}
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\newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}}
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\newcommand{\bv}[2]{\bar{f}_\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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% matrices
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\newcommand{\br}{\bm{r}}
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\newcommand{\bw}{\bm{w}}
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\newcommand{\bG}{\bm{G}}
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\newcommand{\bS}{\bm{S}}
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\newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}}
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\newcommand{\bH}{\bm{H}}
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\newcommand{\bHc}{\bm{H}^\text{c}}
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\newcommand{\bF}[1]{\bm{F}^{#1}}
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\newcommand{\Ex}[1]{\Omega^{#1}}
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\newcommand{\E}[1]{E^{#1}}
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% elements
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\newcommand{\ew}[1]{w_{#1}}
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\newcommand{\eG}[1]{G_{#1}}
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\newcommand{\eS}[1]{S_{#1}}
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\newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}}
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\newcommand{\eHc}[1]{H_{#1}^\text{c}}
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\newcommand{\eF}[2]{F_{#1}^{#2}}
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% Numbers
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\newcommand{\Nel}{N}
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\newcommand{\Nbas}{K}
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% Ao and MO basis
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\newcommand{\MO}[2]{\phi_{#1}^{#2}}
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\newcommand{\cMO}[2]{c_{#1}^{#2}}
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\newcommand{\AO}[1]{\chi_{#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{\SI}{\textcolor{blue}{supplementary material}}
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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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%%%% added by Manu %%%%%
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\newcommand{\manu}[1]{{\textcolor{blue}{ Manu: #1 }} }
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\newcommand{\beq}{\begin{eqnarray}}
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\newcommand{\eeq}{\end{eqnarray}}
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%
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\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
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\newcommand{\bmg}{\bm{\gamma}} % orbital rotation vector
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\newcommand{\bfx}{\bf{x}}
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\newcommand{\bfr}{\bf{r}}
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%%%%
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\begin{document}
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\title{Supplementary Material for ``Weight-dependent local density-functional approximations 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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\author{Emmanuel Fromager}
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\email{fromagere@unistra.fr}
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\affiliation{\LCQ}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{abstract}
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\end{abstract}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Ensemble Hartree--Fock method}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The Hartree--Fock (HF) ensemble energy can be written
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as
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\beq\label{eq:eHF_ener}
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&&E^{\bw}_{\rm
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HF}({\bm\kappa})=
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\sum_{pq}\langle
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\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{pq}
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\nonumber\\
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&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
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\varphi_r(\bmk)\varphi_s(\bmk)\rangle
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%\times
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\sum^M_{K=0}w^{(K)}D^{(K)}_{pr}D^{(K)}_{qs},
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\nonumber\\
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\eeq
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where the one- and antisymmetrized two-electron integrals read,
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\beq
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\langle
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\varphi_p({\bmk})\vert\hat{h}\vert
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\varphi_q({\bmk})\rangle=\int d{\bfx}\;
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\varphi_p({\bmk},{\bfx})\hat{h}\varphi_q({\bmk},{\bfx})
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\eeq
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with $\hat{h}\equiv-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
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ext}(\bfr)$
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and
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\beq
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&&\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
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\varphi_r(\bmk)\varphi_s(\bmk)\rangle=
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\nonumber\\
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&&\int d{\bfx}_1\int d{\bfx}_2\;
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\varphi_p({\bmk},{\bfx}_1)\varphi_q({\bmk},{\bfx}_2)\frac{1}{\vert
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{\bfr}_1-{\bfr}_2\vert}
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\nonumber
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\\
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&&\times\Big[\varphi_r({\bmk},{\bfx}_1)\varphi_s({\bmk},{\bfx}_2)
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-\varphi_s({\bmk},{\bfx}_1)\varphi_r({\bmk},{\bfx}_2)\Big]
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,
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\eeq
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respectively. Note that we use {\it real algebra} and the shorthand
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notation $\int d{\bfx}\equiv\int
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d{\bfr}\sum_{\sigma}$ for integration over space and
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summation over spin.
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% normalization condition
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%$\sum^M_{K=0}w^{(K)}=1$
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%.
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The antihermitian $\bmk\equiv\{\kappa_{pq}\}_{p>q}$ matrix which appears in the integrals controls
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the rotation of the spin-orbitals as follows,
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\beq\label{eq:orb_taylor_expansion}
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&&\varphi_p({\bmk},{\bfx})=\sum_q\left[e^{-{\bmk}}\right]_{qp}\varphi_q({\bfx})
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\nonumber\\
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&&=
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\varphi_p({\bfx})+\sum_{q<p}\kappa_{pq}\varphi_q({\bfx})-\sum_{q>p}\kappa_{qp}\varphi_q({\bfx})
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+\mathcal{O}\left({\bmk}^2\right).
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\eeq
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The (${\bmk}$-independent) one-electron reduced density matrices (1RDMs)
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in Eq.~(\ref{eq:eHF_ener}) are defined in the unrotated molecular
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spin-orbital basis for each (unrotated) determinant $\Phi^{(K)}$ belonging to
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the ensemble as follows: $D^{(K)}_{pr}=\delta_{pr}$ if $\varphi_p$ and
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$\varphi_r$ are both
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occupied in $\Phi^{(K)}$, otherwise $D^{(K)}_{pr}=0$. If the unrotated
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spin-orbitals are the minimizing ensemble HF ones, then the following
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stationarity condition is fulfilled,
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\beq\label{eq:station_cond}
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\left.\dfrac{\partial E^{\bw}_{\rm
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HF}({\bm\kappa})}{\partial \kappa_{lm}}
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\right|_{{\bmk}=0}=0,
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\eeq
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with $l>m$. Since, according to Eq.~(\ref{eq:orb_taylor_expansion}),
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\beq
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\left.\dfrac{\partial
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\varphi_p({\bmk},{\bfx})}{\partial \kappa_{lm}}
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\right|_{{\bmk}=0}=\delta_{lp}\varphi_m({\bfx})-\delta_{mp}\varphi_l({\bfx}),
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\eeq
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Eq.~(\ref{eq:station_cond}) can be written more explicitly as the
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following commutation relation,
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\iffalse%%%%%
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%%%%%% intermediate steps ... %%%%
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\beq
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\sum^M_{K=0}w^{(K)}\sum_qD^{(K)}_{mq}f^{(K)}_{lq}
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-
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\sum^M_{K=0}w^{(K)}\sum_qf^{(K)}_{mq}D^{(K)}_{lq}=0
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\eeq
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% original %%
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\iffalse%%%
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\beq
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&&
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2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
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\nonumber\\
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&&-
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2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
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\nonumber\\
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&&+2\sum_{qrs}\langle \varphi_m\varphi_q\vert\vert
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\varphi_r\varphi_s\rangle
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%\times
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\sum^M_{K=0}w^{(K)}D^{(K)}_{lr}D^{(K)}_{qs}
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\nonumber\\
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&&-2\sum_{qrs}\langle \varphi_l\varphi_q\vert\vert
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\varphi_r\varphi_s\rangle
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%\times
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\sum^M_{K=0}w^{(K)}D^{(K)}_{mr}D^{(K)}_{qs}
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\nonumber\\
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&&=0\eeq
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\fi%%%%
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%%%%%%%
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\beq
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&&
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2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
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\nonumber\\
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&&-
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2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
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\nonumber\\
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&&+2\sum_{qrs}\langle \varphi_m\varphi_r\vert\vert
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\varphi_q\varphi_s\rangle
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%\times
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\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}D^{(K)}_{rs}
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\nonumber\\
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&&-2\sum_{qrs}\langle \varphi_l\varphi_r\vert\vert
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\varphi_q\varphi_s\rangle
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%\times
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\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}D^{(K)}_{rs}
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\nonumber\\
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&&=0\eeq
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%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%
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\fi%%%%
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%%%%%%%
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\beq\label{eq:stat_cond_commut_ind}
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\sum^M_{K=0}w^{(K)}\left[{\bm f}^{(K)},{\bm D}^{(K)}\right]=0,
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\eeq
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where the $K$th Fock matrix elements read
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\beq
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f^{(K)}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
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+
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\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
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\varphi_q\varphi_s\rangle D^{(K)}_{rs}.
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\eeq
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In the minimizing ensemble HF spin-orbital basis, Eq.~(\ref{eq:stat_cond_commut_ind}) reads
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\beq
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\sum^M_{K=0}w^{(K)}\Big(\nu^{(K)}_m-\nu_l^{(K)}\Big)f^{(K)}_{lm}=0,
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\eeq
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where $\nu^{(K)}_m$ is the occupation of the spin-orbital $\varphi_m$ in the
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determinant $\Phi^{(K)}$.\\
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Note that, in more conventional ensemble calculations, the following HF
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energy expression is employed,
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\beq\label{eq:GI_ensHF_ener}
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&&\tilde{E}^{\bw}_{\rm
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HF}({\bm\kappa})=
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\sum_{pq}\langle
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\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle D^{\bw}_{pq}
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\nonumber\\
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&&
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+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
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\varphi_r(\bmk)\varphi_s(\bmk)\rangle
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%\times
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D^{\bw}_{pr}D^{\bw}_{qs},
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\eeq
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where ${\bm D}^{\bw}=\sum^M_{K=0}w^{(K)}{\bm D}^{(K)}$ is the ensemble
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1RDM. In this case, the stationarity condition simply reads
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\beq
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\left[{\bm f}^{\bw},{\bm D}^{\bw}\right]=0,
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\eeq
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where the ensemble Fock matrix elements are defined as follows,
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\beq
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f^{\bw}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
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+
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\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
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\varphi_q\varphi_s\rangle D^{\bw}_{rs}.
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\eeq
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The major issue with the expression of the ensemble energy in
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Eq.~(\ref{eq:GI_ensHF_ener}) is the
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ghost-interaction error from which our expression (see
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Eq.~(\ref{eq:eHF_ener})) is free. Note also
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that, by construction, the ensemble energy in Eq.~(\ref{eq:GI_ensHF_ener}) is quadratic in the
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ensemble weights while ours, like the exact one, varies linearly with
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the weights.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Ensemble Hartree--Fock exchange and density-functional
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ghost-interaction correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\beq
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F^{\bw}_{\rm HF}[n]&=&
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\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
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HF}\left[{\bmg}^{\bw}\right]\right\}
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\nonumber\\
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&=&{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
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HF}\left[{\bmg}^{\bw}[n]\right]
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\eeq
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where
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$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\gamma}^{(K)}$ is an ensemble density matrix operator constructed
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from Slater determinants, the ensemble 1RDM elements are $\gamma_{pq}^{\bw}={\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
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and $W_{\rm
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HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
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\varphi_r\varphi_s\rangle
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%\times
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\gamma_{pr}\gamma_{qs}$.\\
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In-principle-exact decomposition:
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\beq
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F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
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Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
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\eeq
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The complementary ensemble Hx energy removes the ghost-interaction
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errors introduced in $W_{\rm
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HF}\left[{\bmg}^{\bw}[n]\right]$:
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\beq
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\overline{E}^{{\bw}}_{\rm
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Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
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HF}\left[{\bmg}^{(K)}[n]\right]
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-W_{\rm
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HF}\left[{\bmg}^{\bw}[n]\right],
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\eeq
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which gives in the canonical orbital basis
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\beq
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&&\overline{E}^{{\bw}}_{\rm
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Hx}[n]=
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\dfrac{1}{2}\sum_{pq}
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\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
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\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
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\nonumber\\
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&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
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-\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
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.\eeq
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\manu{I would guess that, in a uniform system, the GOK-DFT and our
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canonical orbitals are the same. This is nice since we can construct
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in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
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Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
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Variational expression for the ensemble energy:
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\beq
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E^{{\bw}}=\underset{\hat{\gamma}^{{\bw}}}{\rm min}\Big\{
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&&{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
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HF}\left[{\bmg}^{\bw}\right]
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+
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\overline{E}^{{\bw}}_{\rm
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Hxc}\left[n_{\hat{\gamma}^{{\bw}}}\right]
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%+E^{{\bw}}_{\rm c}\left[n_{\hat{\gamma}^{{\bw}}}\right]
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\nonumber\\
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&&
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+\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\gamma}^{{\bw}}}({\bfr})
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\Big\}
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\eeq
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Note that, if we use orbital rotations, the gradient of the DFT energy
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contributions can be expressed as follows,
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\beq
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\left.\dfrac{\partial
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\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
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}{\partial \kappa_{lm}}
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\right|_{{\bmk}=0}=\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
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|
Hxc}\left[n^{{\bw}}\right]}{\delta
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n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}}
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\right|_{{\bmk}=0},
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\eeq
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where
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\beq
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n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw}
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\eeq
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thus leading to
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\beq
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&&\left.\dfrac{\partial
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\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
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}{\partial \kappa_{lm}}
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\right|_{{\bmk}=0}=
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\sum_{pq}\gamma_{pq}^{\bw}
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\nonumber\\
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&&\times\left.\dfrac{\partial}
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{\partial \kappa_{lm}}
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\Big[\left\langle\varphi_p(\bmk)\middle\vert\hat{\overline{v}}^{{\bw}}_{\rm
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Hxc}
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\middle\vert \varphi_q(\bmk)\right\rangle
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\Big]
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\right|_{{\bmk}=0}.
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\eeq
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In conclusion, the minimizing canonical orbitals fulfill the following
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hybrid HF/GOK-DFT equation,
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\beq
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&&\left(-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
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ext}({\bfr})+\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]
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+\dfrac{\delta \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]}{\delta
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n({\br})}\right)\varphi^{{\bw}}_p({\bfx})
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\nonumber
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\\
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&&=\varepsilon^{{\bw}}_p\varphi^{{\bw}}_p({\bfx}).
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\eeq
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Since $\partial \gamma_{pq}^{\bw}/\partial
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w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
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\manu{just for me ...
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\beq
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&&+\dfrac{1}{2}
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\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
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\varphi_r\varphi_s\rangle
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%\times
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\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
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\nonumber\\
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&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_q\varphi_p\vert\vert
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\varphi_s\varphi_r\rangle
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%\times
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\gamma^{\bw}_{pr}\left(\gamma_{qs}^{(I)}-\gamma_{qs}^{(0)}\right)
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\nonumber\\
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&&=
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\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
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\varphi_r\varphi_s\rangle
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%\times
|
|
\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
|
|
\nonumber\\
|
|
&&=
|
|
\sum_{pr}\left[\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]\right]_{pr}\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)
|
|
\nonumber\\
|
|
&&=
|
|
\sum_p\left[\hat{u}_{\rm
|
|
HF}\left[\gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right)
|
|
\eeq
|
|
}
|
|
|
|
\beq
|
|
\dfrac{dE^{\bw}}{dw^{(I)}}=\sum_p\varepsilon^{{\bw}}_p\left(\nu_p^{(I)}-\nu_p^{(0)}\right)+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
|
|
Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
|
|
\eeq
|
|
|
|
LZ shift in this context: $\varepsilon^{{\bw}}_p\rightarrow
|
|
\overline{\varepsilon}^{{\bw}}_p=\varepsilon^{{\bw}}_p+\overline{\Delta}_{\rm
|
|
LZ}^{{\bw}}$ where
|
|
|
|
\beq
|
|
N\overline{\Delta}_{\rm
|
|
LZ}^{{\bw}}&=&\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]
|
|
-\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
|
|
Hxc}\left[n^{{\bw}}\right]}{\delta
|
|
n({\br})}n^{{\bw}}({\bfr})
|
|
\nonumber\\
|
|
&&
|
|
-W_{\rm
|
|
HF}\left[{\bmg}^{\bw}\right]
|
|
\eeq
|
|
|
|
such that
|
|
\beq
|
|
E^{{\bw}}=\sum^M_{K=0}w^{(K)}\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p.
|
|
\eeq
|
|
|
|
Thus we conclude that individual energies can be expressed in principle
|
|
exactly as follows,
|
|
|
|
\beq
|
|
E^{(K)}=\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p+\sum^M_{I>0}\left(\delta_{IK}-w^{(I)}\right)\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
|
|
Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
|
|
\eeq
|
|
%%%%%%%%%%%%%%
|
|
\iffalse%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Generalized GOK-DFT}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
The energy to be minimized in a generalized GOK-DFT approach can be
|
|
written as
|
|
\beq
|
|
\eeq
|
|
\fi%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Construction of the density-functional approximations}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
The density-functional approximations designed in this manuscript are based on highly-accurate energies 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.
|
|
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
|
|
|
|
The reduced (i.e.~per electron) HF energy for these three states is:
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\e{HF}{(0)}(n) & = \frac{\pi^2}{8} n^2 + n,
|
|
\\
|
|
\e{HF}{(1)}(n) & = \frac{\pi^2}{2}n^2 + \frac{4}{3} n,
|
|
\\
|
|
\e{HF}{(2)}(n) & = \frac{9\pi^2}{8}n^2 + \frac{23}{15} n.
|
|
\end{align}
|
|
\end{subequations}
|
|
All these states have the same (uniform) density $n = 2/(2\pi R)$ where $R$ is the radius of the ring on which the electrons are confined.
|
|
|
|
The total energy of the ground and doubly-excited states are given by the two lowest eigenvalues of the Hamiltonian $\bH$ with elements
|
|
\begin{equation}
|
|
\begin{split}
|
|
H_{ij}
|
|
& = \int_0^\pi \qty[ \frac{\psi_i(\omega)}{R} \frac{\psi_j(\omega)}{R} + \frac{\psi_i(\omega)\psi_j(\omega)}{2R\sin(\omega/2)} ] d\omega
|
|
\\
|
|
& = \frac{\sqrt{\pi}}{2 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{2})} + \frac{ij}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+2}{2})} ],
|
|
\end{split}
|
|
\end{equation}
|
|
where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function, \cite{NISTbook} and
|
|
\begin{equation}
|
|
\psi_i(\omega) = \sin(\omega/2) \sin^{i-1}(\omega/2), \quad i=1,\ldots,M
|
|
\end{equation}
|
|
are (non-orthogonal) explicitly-correlated basis functions with overlap matrix elements
|
|
\begin{equation}
|
|
S_{ij}
|
|
= \int_0^\pi \psi_i(\omega)\psi_j(\omega) d\omega
|
|
= \sqrt{\pi} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+2}{2})}.
|
|
\end{equation}
|
|
Thanks to this explicitly-correlated basis, the convergence rate of the energy is exponential with respect to $M$.
|
|
Therefore, high accuracy is reached with a very small number of basis functions.
|
|
Here, we typically use $M=10$.
|
|
For the singly-excited state, one has to modify the basis functions as
|
|
\begin{equation}
|
|
\psi_i(\omega) = \cos(\omega/2) \sin^{i-1} (\omega/2),
|
|
\end{equation}
|
|
and its energy is obtained by the lowest root of the Hamiltonian in this basis, and the matrix elements reads
|
|
\begin{align}
|
|
H_{ij} & = \frac{\sqrt{\pi}}{4 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{3})} + \frac{3ij+i+j-1}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+4}{2})} ],
|
|
\\
|
|
S_{ij} & = \frac{\sqrt{\pi}}{2} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+4}{2})}.
|
|
\end{align}
|
|
The numerical values of the correlation energy for various $R$ are reported in Table \ref{tab:Ref} for the three states of interest.
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{Ec}
|
|
\caption{
|
|
$\e{c}{(I)}$ [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}
|
|
\end{figure}
|
|
%%% %%% %%%
|
|
|
|
%%% TABLE I %%%
|
|
\begin{turnpage}
|
|
\begin{squeezetable}
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:Ref}
|
|
$-\e{c}{(I)}$ 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}{lcddddddddddd}
|
|
State & $I$ & \mc{11}{c}{Ring's radius $R = 1/(\pi n)$} \\
|
|
\cline{3-13}
|
|
& & \tabc{$0$} & \tabc{$1/10$} & \tabc{$1/5$} & \tabc{$1/2$} & \tabc{$1$} & \tabc{$2$} & \tabc{$5$} & \tabc{$10$} & \tabc{$20$} & \tabc{$50$} & \tabc{$100$} \\
|
|
\hline
|
|
Ground state & $0$ & 0.013708 & 0.012859 & 0.012525 & 0.011620 & 0.010374 & 0.008558 & 0.005673 & 0.003697 & 0.002226 & 0.001046 & 0.000567 \\
|
|
Singly-excited state & $1$ & 0.0238184 & 0.023392 & 0.022979 & 0.021817 & 0.020109 & 0.017371 & 0.012359 & 0.008436 & 0.005257 & 0.002546 & 0.001399 \\
|
|
Doubly-excited state & $2$ & 0.018715 & 0.018653 & 0.018576 & 0.018300 & 0.017743 & 0.016491 & 0.013145 & 0.009670 & 0.006365 & 0.003231 & 0.001816 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
\end{squeezetable}
|
|
\end{turnpage}
|
|
|
|
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)}},
|
|
\end{equation}
|
|
where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
|
|
The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
|
Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
|
|
|
|
|
|
%%% FIG 2 %%%
|
|
\begin{figure*}
|
|
\includegraphics[height=0.25\linewidth]{EvsL_2}
|
|
\includegraphics[height=0.25\linewidth]{EvsL_3}
|
|
\includegraphics[height=0.25\linewidth]{EvsL_4}
|
|
\includegraphics[height=0.25\linewidth]{EvsL_5}
|
|
\includegraphics[height=0.25\linewidth]{EvsL_6}
|
|
\includegraphics[height=0.25\linewidth]{EvsL_7}
|
|
\caption{
|
|
Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the box length $L$ for various methods.
|
|
}
|
|
\label{fig:EvsL}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure*}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_0125}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_025}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_05}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_1}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_2}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_4}
|
|
\includegraphics[height=0.25\linewidth]{EvsN_8}
|
|
\caption{
|
|
Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the number of electrons $N$ for various methods and box length $L$.
|
|
}
|
|
\label{fig:EvsL}
|
|
\end{figure*}
|
|
%%% %%% %%%
|
|
|
|
|
|
%%% TABLE I %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OptGap}
|
|
Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
|
|
The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lclddddddd}
|
|
& & & \mc{7}{c}{$L/\pi$} \\
|
|
\cline{4-10}
|
|
Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
|
|
\hline
|
|
FCI & & $\E{(0)}$ & 168.1946 & 44.0662 & 12.0035 & 3.4747 & 1.0896 & 0.3719 & 0.1367 \\
|
|
& & $\E{(1)}$ & 330.2471 & 85.0890 & 22.5112 & 6.2247 & 1.8355 & 0.5845 & 0.2006 \\
|
|
& & $\E{(2)}$ & 809.9972 & 204.9840 & 52.4777 & 13.7252 & 3.7248 & 1.0696 & 0.3300 \\
|
|
& & $\Ex{(1)}$ & 162.0525 & 41.0228 & 10.5078 & 2.7500 & 0.7458 & 0.2125 & 0.0639 \\
|
|
& & $\Ex{(2)}$ & 641.8026 & 160.9177 & 40.4743 & 10.2505 & 2.6352 & 0.6977 & 0.1933 \\
|
|
\\
|
|
CIS & & $\Ex{(1)}$ & 0.0104 & 0.0102 & 0.0099 & 0.0092 & 0.0077 & 0.0051 & 0.0022 \\
|
|
\\
|
|
TDHF & & $\Ex{(1)}$ & 0.0019 & 0.0021 & 0.0023 & 0.0027 & 0.0029 & 0.0023 & 0.0011 \\
|
|
\\
|
|
TDA-TDLDA& & $\Ex{(1)}$ & 0.0099 & 0.0088 & 0.0058 & -0.0041 & -0.0316 & -0.0467 & \\
|
|
\\
|
|
TDLDA & & $\Ex{(1)}$ & 0.0015 & 0.0006 & -0.0018 & -0.0106 & -0.0370 & -0.0518 & \\
|
|
\\
|
|
eDFT & $(0,0)$ & $\E{(0)}$ & -0.0397 & -0.0391 & -0.0380 & -0.0361 & -0.0323 & -0.0236 \\
|
|
& & $\E{(1)}$ & 0.0215 & 0.0213 & 0.0210 & 0.0200 & 0.0159 & 0.0102 \\
|
|
& & $\E{(2)}$ & -0.0426 & -0.0425 & -0.0419 & -0.0386 & -0.0249 & -0.0044 \\
|
|
& & $\Ex{(1)}$ & 0.0612 & 0.0604 & 0.0590 & 0.0561 & 0.0483 & 0.0338 \\
|
|
& & $\Ex{(2)}$ & -0.0029 & -0.0034 & -0.0039 & -0.0025 & 0.0074 & 0.0191 \\
|
|
& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
|
|
& & $\DD{c}{(1)}$ & 0.0064 & 0.0056 & 0.0043 & 0.0022 & -0.0007 & -0.0037 & \\
|
|
& & $\DD{c}{(2)}$ & 0.0159 & 0.0147 & 0.0126 & 0.0093 & 0.0046 & -0.0009 & \\
|
|
& & $\LZ{c}{}$ & 0.0013 & 0.0024 & 0.0043 & 0.0070 & 0.0100 & 0.0121 & \\
|
|
\\
|
|
eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0031 & 0.0036 & 0.0044 & 0.0054 & 0.0042 & -0.0025 \\
|
|
& & $\E{(1)}$ & 0.0090 & 0.0087 & 0.0083 & 0.0076 & 0.0070 & 0.0071 \\
|
|
& & $\E{(2)}$ & -0.0005 & -0.0009 & -0.0015 & -0.0023 & -0.0030 & -0.0026 \\
|
|
& & $\Ex{(1)}$ & 0.0058 & 0.0052 & 0.0039 & 0.0022 & 0.0028 & 0.0096 \\
|
|
& & $\Ex{(2)}$ & -0.0036 & -0.0045 & -0.0058 & -0.0077 & -0.0072 & 0.0000 \\
|
|
& & $\DD{c}{(0)}$ & -0.0074 & -0.0067 & -0.0055 & -0.0036 & -0.0010 & 0.0019 & \\
|
|
& & $\DD{c}{(1)}$ & -0.0010 & -0.0011 & -0.0014 & -0.0017 & -0.0021 & -0.0022& \\
|
|
& & $\DD{c}{(2)}$ & 0.0084 & 0.0079 & 0.0069 & 0.0053 & 0.0031 & 0.0003 & \\
|
|
& & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0023 & 0.0040 & 0.0063 & 0.0087 & \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
|
|
%%% TABLE II %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OptGap}
|
|
Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
|
|
The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lclddddddd}
|
|
& & & \mc{7}{c}{$L/\pi$} \\
|
|
\cline{4-10}
|
|
Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
|
|
\hline
|
|
FCI & & $\E{(0)}$ & 475.6891 & 125.7776 & 34.8248 & 10.3536 & 3.3766 & 1.2126 & 0.4721 \\
|
|
& & $\E{(1)}$ & 702.8330 & 183.3370 & 49.5922 & 14.2255 & 4.4269 & 1.5105 & 0.5606 \\
|
|
& & $\E{(2)}$ & 1379.3128 & 353.5967 & 92.7398 & 25.3135 & 7.3546 & 2.3203 & 0.7990 \\
|
|
& & $\Ex{(1)}$ & 227.1438 & 57.5594 & 14.7674 & 3.8720 & 1.0504 & 0.2979 & 0.0885 \\
|
|
& & $\Ex{(2)}$ & 903.6236 & 227.8191 & 57.9150 & 14.9599 & 3.9780 & 1.1077 & 0.3269 \\
|
|
\\
|
|
CIS & & $\Ex{(1)}$ & 0.0163 & 0.0161 & 0.0157 & 0.0149 & 0.0133 & 0.0102 & 0.0057 \\
|
|
\\
|
|
TDHF & & $\Ex{(1)}$ & 0.0013 & 0.0013 & 0.0014 & 0.0014 & 0.0013 & 0.0010 & 0.0007 \\
|
|
\\
|
|
TDA-TDLDA& & $\Ex{(1)}$ & 0.0162 & 0.0157 & 0.0146 & 0.0110 & -0.0049 & -0.0344 & -0.0378 \\
|
|
\\
|
|
TDLDA & & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0264 & 0.0269 & 0.0273 & 0.0206 & -0.0116 \\
|
|
\\
|
|
eDFT & $(0,0)$ & $\E{(0)}$ & -0.0481 & -0.0478 & -0.0473 & -0.0463 & -0.0446 & -0.0387 & -0.0257 \\
|
|
& & $\E{(1)}$ & 0.0343 & 0.0336 & 0.0321 & 0.0292 & 0.0220 & 0.0084 & 0.0008 \\
|
|
& & $\E{(2)}$ & 0.0277 & 0.0267 & 0.0247 & 0.0216 & 0.0187 & 0.0208 & 0.0209 \\
|
|
& & $\Ex{(1)}$ & 0.0824 & 0.0814 & 0.0794 & 0.0755 & 0.0666 & 0.0471 & 0.0266 \\
|
|
& & $\Ex{(2)}$ & 0.0759 & 0.0745 & 0.0720 & 0.0679 & 0.0633 & 0.0595 & 0.0467 \\
|
|
& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
|
|
& & $\DD{c}{(1)}$ & 0.0100 & 0.0092 & 0.0077 & 0.0051 & 0.0012 & -0.0034 & -0.0072 \\
|
|
& & $\DD{c}{(2)}$ & 0.0244 & 0.0231 & 0.0208 & 0.0168 & 0.0108 & 0.0029 & -0.0050 \\
|
|
& & $\LZ{c}{}$ & 0.0014 & 0.0026 & 0.0048 & 0.0083 & 0.0127 & 0.0169 & 0.01860 \\
|
|
\\
|
|
eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0078 & 0.0080 & 0.0082 & 0.0085 & 0.0081 & 0.0024 & -0.0022 \\
|
|
& & $\E{(1)}$ & 0.0172 & 0.0162 & 0.0144 & 0.0112 & 0.0064 & 0.0019 & 0.0004 \\
|
|
& & $\E{(2)}$ & 0.0645 & 0.0636 & 0.0621 & 0.0590 & 0.0530 & 0.0420 & 0.0300 \\
|
|
& & $\Ex{(1)}$ & 0.0094 & 0.0083 & 0.0062 & 0.0027 & -0.0018 & -0.0004 & 0.0026 \\
|
|
& & $\Ex{(2)}$ & 0.0567 & 0.0557 & 0.0539 & 0.0506 & 0.0449 & 0.0397 & 0.0323 \\
|
|
& & $\DD{c}{(0)}$ & -0.0115 & -0.0107 & -0.0094 & -0.0072 & -0.0038 & 0.0005 & 0.0045 \\
|
|
& & $\DD{c}{(1)}$ & -0.0015 & -0.0016 & -0.0018 & -0.0022 & -0.0028 & -0.0033 & -0.0032 \\
|
|
& & $\DD{c}{(2)}$ & 0.0129 & 0.0123 & 0.0113 & 0.0094 & 0.0066 & 0.0028 & -0.0013 \\
|
|
& & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0025 & 0.0044 & 0.0074 & 0.0110 & 0.0143 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
|
|
|
|
%%% TABLE III %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OptGap}
|
|
Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
|
|
The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lclddddddd}
|
|
& & & \mc{7}{c}{$L/\pi$} \\
|
|
\cline{4-10}
|
|
Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
|
|
\hline
|
|
FCI & & $\E{(0)}$ & 1020.3778 & 270.0849 & 74.9426 & 22.3790 & 7.3595 & 2.6798 & 1.0633 \\
|
|
& & $\E{(1)}$ & 1312.2776 & 344.0184 & 93.8936 & 27.3398 & 8.7021 & 3.0600 & 1.1764 \\
|
|
& & $\E{(2)}$ & 2183.4399 & 563.5949 & 149.6753 & 41.7213 & 12.5052 & 4.1033 & 1.4749 \\
|
|
& & $\Ex{(1)}$ & 291.8998 & 73.9335 & 18.9510 & 4.9608 & 1.3426 & 0.3802 & 0.1131 \\
|
|
& & $\Ex{(2)}$ & 1163.0621 & 293.5099 & 74.7326 & 19.3423 & 5.1457 & 1.4235 & 0.4116 \\
|
|
\\
|
|
CIS & & $\Ex{(1)}$ & 0.0203 & 0.0202 & 0.0200 & 0.0195 & 0.0187 & 0.0167 & 0.0116 \\
|
|
\\
|
|
TDHF & & $\Ex{(1)}$ & 0.0008 & 0.0008 & 0.0009 & 0.0009 & 0.0008 & 0.0008 & 0.0007 \\
|
|
\\
|
|
TDA-TDLDA& & $\Ex{(1)}$ & 0.0203 & 0.0201 & 0.0195 & 0.0181 & 0.0106 & -0.0178 & -0.0369 \\
|
|
\\
|
|
TDLDA & & $\Ex{(1)}$ & 0.0008 & 0.0007 & 0.0004 & -0.0006 & -0.0074 & -0.0360 & -0.0653 \\
|
|
\\
|
|
eDFT & $(0,0)$ & $\E{(0)}$ & -0.0541 & -0.0539 & -0.0537 & -0.0534 & -0.0529 & -0.0504 & -0.0386 \\
|
|
& & $\E{(1)}$ & 0.0413 & 0.0406 & 0.0390 & 0.0362 & 0.0304 & 0.0159 & 0.0008 \\
|
|
& & $\E{(2)}$ & 0.0642 & 0.0622 & 0.0586 & 0.0517 & 0.0399 & 0.0254 & 0.0149 \\
|
|
& & $\Ex{(1)}$ & 0.0954 & 0.0945 & 0.0927 & 0.0896 & 0.0833 & 0.0663 & 0.0394 \\
|
|
& & $\Ex{(2)}$ & 0.1182 & 0.1162 & 0.1123 & 0.1051 & 0.0928 & 0.0758 & 0.0534 \\
|
|
& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
|
|
& & $\DD{c}{(1)}$ & 0.0136 & 0.0127 & 0.0111 & 0.0083 & 0.0038 & -0.0022 & -0.0080 \\
|
|
& & $\DD{c}{(2)}$ & 0.0330 & 0.0316 & 0.0291 & 0.0248 & 0.0178 & 0.0080 & -0.0028 \\
|
|
& & $\LZ{c}{}$ & 0.0014 & 0.0027 & 0.0051 & 0.0091 & 0.0147 & 0.0207 & 0.0245 \\
|
|
\\
|
|
eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0085 & 0.0085 & 0.0084 & 0.0082 & 0.0072 & 0.0021 & -0.0015 \\
|
|
& & $\E{(1)}$ & 0.0164 & 0.0152 & 0.0129 & 0.0087 & 0.0020 & -0.0050 & -0.0044 \\
|
|
& & $\E{(2)}$ & 0.0936 & 0.0917 & 0.0880 & 0.0807 & 0.0664 & 0.0434 & 0.0300 \\
|
|
& & $\Ex{(1)}$ & 0.0079 & 0.0067 & 0.0045 & 0.0006 & -0.0051 & -0.0071 & -0.0029 \\
|
|
& & $\Ex{(2)}$ & 0.0851 & 0.0832 & 0.0796 & 0.0725 & 0.0593 & 0.0413 & 0.0315 \\
|
|
& & $\DD{c}{(0)}$ & -0.0155 & -0.0148 & -0.0134 & -0.0110 & -0.0071 & -0.0017 & 0.0040 \\
|
|
& & $\DD{c}{(1)}$ & -0.0020 & -0.0021 & -0.0023 & -0.0027 & -0.0034 & -0.0042 & -0.0044 \\
|
|
& & $\DD{c}{(2)}$ & 0.0175 & 0.0168 & 0.0157 & 0.0137 & 0.0105 & 0.0059 & 0.0004 \\
|
|
& & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0025 & 0.0047 & 0.0081 & 0.0127 & 0.0175 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
|
|
%%% TABLE IV %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OptGap}
|
|
Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
|
|
The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lclddddddd}
|
|
& & & \mc{7}{c}{$L/\pi$} \\
|
|
\cline{4-10}
|
|
Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
|
|
\hline
|
|
FCI & & $\E{(0)}$ & 1867.6344 & 493.6760 & 136.7020 & 40.7244 & 13.3763 & 4.8811 & 1.9492 \\
|
|
& & $\E{(1)}$ & 2224.11488 & 583.8981 & 159.7957 & 46.7553 & 15.0029 & 5.3399 & 2.0855 \\
|
|
& & $\E{(2)}$ & 3289.2022 & 852.4249 & 228.0415 & 64.3597 & 19.6613 & 6.6206 & 2.4547 \\
|
|
& & $\Ex{(1)}$ & 356.4804 & 90.2221 & 23.0937 & 6.0308 & 1.6266 & 0.4588 & 0.1363 \\
|
|
& & $\Ex{(2)}$ & 1421.56773 & 358.7489 & 91.3395 & 23.6352 & 6.2850 & 1.7395 & 0.5055 \\
|
|
\\
|
|
CIS & & $\Ex{(1)}$ & 0.0230 & 0.0230 & 0.0229 & 0.0229 & 0.0230 & 0.0225 & 0.0182 \\
|
|
\\
|
|
TDHF & & $\Ex{(1)}$ & 0.0005 & 0.0005 & 0.0005 & 0.0005 & 0.0005 & 0.0005 & 0.0006 \\
|
|
\\
|
|
TDA-TDLDA& & $\Ex{(1)}$ & 0.0230 & 0.0230 & 0.0228 & 0.0223 & 0.0192 & -0.0015 & -0.0309 \\
|
|
\\
|
|
TDLDA & & $\Ex{(1)}$ & 0.0005 & 0.0005 & 0.0004 & 0.0000 & -0.0033 & -0.0248 & -0.0650 \\
|
|
\\
|
|
eDFT & $(0,0)$ & $\E{(0)}$ & -0.0587 & -0.0586 & -0.0587 & -0.0588 & -0.0591 & -0.0590 & -0.0506 \\
|
|
& & $\E{(1)}$ & 0.0457 & 0.0450 & 0.0435 & 0.0409 & 0.0362 & 0.0241 & 0.0033 \\
|
|
& & $\E{(2)}$ & 0.0861 & 0.0838 & 0.0793 & 0.0712 & 0.0571 & 0.0377 & 0.0196 \\
|
|
& & $\Ex{(1)}$ & 0.1044 & 0.1036 & 0.1022 & 0.0997 & 0.0953 & 0.0830 & 0.0540 \\
|
|
& & $\Ex{(2)}$ & 0.1447 & 0.1424 & 0.1380 & 0.1300 & 0.1162 & 0.0966 & 0.0703 \\
|
|
& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
|
|
& & $\DD{c}{(1)}$ & 0.0172 & 0.0163 & 0.0147 & 0.0117 & 0.0067 & -0.0004 & -0.0080 \\
|
|
& & $\DD{c}{(2)}$ & 0.0416 & 0.0402 & 0.0376 & 0.0329 & 0.0253 & 0.0140 & 0.0005 \\
|
|
& & $\LZ{c}{}$ & 0.0015 & 0.0028 & 0.0053 & 0.0096 & 0.0161 & 0.0238 & 0.0297 \\
|
|
\\
|
|
eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0070 & 0.0070 & 0.0068 & 0.0063 & 0.0053 & 0.0015 & -0.0049 \\
|
|
& & $\E{(1)}$ & 0.0162 & 0.0151 & 0.0128 & 0.0086 & 0.0018 & -0.0066 & -0.0095 \\
|
|
& & $\E{(2)}$ & 0.1080 & 0.1056 & 0.1011 & 0.0925 & 0.0772 & 0.0538 & 0.0325 \\
|
|
& & $\Ex{(1)}$ & 0.0092 & 0.0081 & 0.0060 & 0.0022 & -0.0035 & -0.0081 & -0.0047 \\
|
|
& & $\Ex{(2)}$ & 0.1010 & 0.0986 & 0.0943 & 0.0862 & 0.0719 & 0.0523 & 0.0373 \\
|
|
& & $\DD{c}{(0)}$ & -0.0196 & -0.0188 & -0.0174 & -0.0148 & -0.0106 & -0.0044 & 0.0029 \\
|
|
& & $\DD{c}{(1)}$ & -0.0024 & -0.0025 & -0.0027 & -0.0032 & -0.0040 & -0.0050 & -0.0056 \\
|
|
& & $\DD{c}{(2)}$ & 0.0220 & 0.0213 & 0.0201 & 0.0180 & 0.0146 & 0.0093 & 0.0027 \\
|
|
& & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0026 & 0.0049 & 0.0087 & 0.0141 & 0.0202 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
|
|
%%% TABLE V %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OptGap}
|
|
Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
|
|
The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lclddddddd}
|
|
& & & \mc{7}{c}{$L/\pi$} \\
|
|
\cline{4-10}
|
|
Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
|
|
\hline
|
|
FCI & & $\E{(0)}$ & 3082.5386 & 813.0910 & 224.3734 & 66.5257 & 21.7454 & 7.9136 & 3.1633 \\
|
|
& & $\E{(1)}$ & 3503.4911 & 919.5487 & 251.5842 & 73.6145 & 23.6504 & 8.4487 & 3.3217 \\
|
|
& & $\E{(2)}$ & 4762.0921 & 1236.8257 & 332.1993 & 94.3988 & 29.1455 & 9.9582 & 3.7572 \\
|
|
& & $\Ex{(1)}$ & 420.9525 & 106.4577 & 27.2108 & 7.0888 & 1.9050 & 0.5351 & 0.1583 \\
|
|
& & $\Ex{(2)}$ & 1679.5536 & 423.7347 & 107.8259 & 27.8731 & 7.4001 & 2.0446 & 0.5938 \\
|
|
\\
|
|
CIS & & $\Ex{(1)}$ & 0.0249 & 0.0248 & 0.0250 & 0.0253 & 0.0261 & 0.0272 & 0.0248 \\
|
|
\\
|
|
TDHF & & $\Ex{(1)}$ & 0.0002 & 0.0000 & 0.0003 & 0.0003 & 0.0003 & 0.0003 & 0.0004 \\
|
|
\\
|
|
TDA-TDLDA& & $\Ex{(1)}$ & 0.0249 & 0.0248 & 0.0250 & 0.0250 & 0.0242 & 0.0114 & -0.0223 \\
|
|
\\
|
|
TDLDA & & $\Ex{(1)}$ & 0.0002 & 0.0000 & 0.0002 & 0.0000 & -0.0016 & -0.0162 & -0.0612 \\
|
|
\\
|
|
eDFT & $(0,0)$ & $\E{(0)}$ & -0.0626 & -0.0627 & -0.0628 & -0.0632 & -0.0641 & -0.0654 & -0.0612 \\
|
|
& & $\E{(1)}$ & 0.0486 & 0.0477 & 0.0465 & 0.0440 & 0.0400 & 0.0308 & 0.0078 \\
|
|
& & $\E{(2)}$ & 0.1017 & 0.0992 & 0.0946 & 0.0862 & 0.0718 & 0.0507 & 0.0271 \\
|
|
& & $\Ex{(1)}$ & 0.1112 & 0.1104 & 0.1093 & 0.1072 & 0.1041 & 0.0962 & 0.0690 \\
|
|
& & $\Ex{(2)}$ & 0.1643 & 0.1619 & 0.1575 & 0.1494 & 0.1358 & 0.1162 & 0.0884 \\
|
|
& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
|
|
& & $\DD{c}{(1)}$ & 0.0208 & 0.0199 & 0.0182 & 0.0151 & 0.0098 & 0.0018 & -0.0075 \\
|
|
& & $\DD{c}{(2)}$ & 0.0503 & 0.0488 & 0.0460 & 0.0412 & 0.0330 & 0.0205 & 0.0046 \\
|
|
& & $\LZ{c}{}$ & 0.0015 & 0.0029 & 0.0054 & 0.0100 & 0.0172 & 0.0264 & 0.0344 \\
|
|
\\
|
|
eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0046 & 0.0045 & 0.0043 & 0.0039 & 0.0031 & 0.0006 & -0.0067 \\
|
|
& & $\E{(1)}$ & 0.0157 & 0.0144 & 0.0123 & 0.0080 & 0.0009 & -0.0091 & -0.0160 \\
|
|
& & $\E{(2)}$ & 0.1167 & 0.1142 & 0.1095 & 0.1007 & 0.0853 & 0.0616 & 0.0355 \\
|
|
& & $\Ex{(1)}$ & 0.0112 & 0.0099 & 0.0080 & 0.0041 & -0.0022 & -0.0097 & -0.0093 \\
|
|
& & $\Ex{(2)}$ & 0.1121 & 0.1097 & 0.1051 & 0.0968 & 0.0822 & 0.0610 & 0.0423 \\
|
|
& & $\DD{c}{(0)}$ & -0.0237 & -0.0229 & -0.0214 & -0.0188 & -0.0142 & -0.0073 & 0.0013 \\
|
|
& & $\DD{c}{(1)}$ & -0.0029 & -0.0030 & -0.0032 & -0.0037 & -0.0045 & -0.0057 & -0.0066 \\
|
|
& & $\DD{c}{(2)}$ & 0.0266 & 0.0259 & 0.0246 & 0.0224 & 0.0187 & 0.0130 & 0.0053 \\
|
|
& & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0026 & 0.0050 & 0.0091 & 0.0151 & 0.0224 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
|
|
|
|
%%% TABLE VI %%%
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:OptGap}
|
|
Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
|
|
The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lclddddddd}
|
|
& & & \mc{7}{c}{$L/\pi$} \\
|
|
\cline{4-10}
|
|
Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
|
|
\hline
|
|
FCI & & $\E{(0)}$ & 4729.98018 & 1244.7753 & 342.1796 & 100.8943 & 32.7728 & 11.8683 & 4.7359 \\
|
|
& & $\E{(1)}$ & 5215.3307 & 1367.4316 & 373.4897 & 109.0326 & 34.9524 & 12.4779 & 4.9156 \\
|
|
& & $\E{(2)}$ & 6667.18516 & 1733.3319 & 466.4133 & 132.9686 & 41.2715 & 14.2096 & 5.4146 \\
|
|
& & $\Ex{(1)}$ & 485.3505 & 122.6563 & 31.3101 & 8.1382 & 2.1796 & 0.6096 & 0.1797 \\
|
|
& & $\Ex{(2)}$ & 1937.2050 & 488.5566 & 124.2336 & 32.0743 & 8.4987 & 2.3413 & 0.6787 \\
|
|
\\
|
|
CIS & & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0265 & 0.0270 & 0.0283 & 0.0308 & 0.0309 \\
|
|
\\
|
|
TDHF & & $\Ex{(1)}$ & 0.0000 & 0.0001 & 0.0000 & 0.0000 & 0.0000 & 0.0001 & 0.0003 \\
|
|
\\
|
|
TDA-TDLDA& & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0264 & 0.0269 & 0.0273 & 0.0206 & -0.0116 \\
|
|
\\
|
|
TDLDA & & $\Ex{(1)}$ & 0.0000 & 0.0001 & 0.0000 & -0.0001 & -0.0009 & -0.0107 & -0.0539 \\
|
|
\\
|
|
eDFT & $(0,0)$ & $\E{(0)}$ & -0.0664 & -0.0666 & -0.0667 & -0.0672 & -0.0684 & -0.0707 & -0.0702 \\
|
|
& & $\E{(1)}$ & 0.0502 & 0.0495 & 0.0482 & 0.0459 & 0.0423 & 0.0355 & 0.0131 \\
|
|
& & $\E{(2)}$ & 0.1122 & 0.1104 & 0.1061 & 0.0979 & 0.0836 & 0.0635 & 0.0360 \\
|
|
& & $\Ex{(1)}$ & 0.1165 & 0.1161 & 0.1149 & 0.1131 & 0.1108 & 0.1062 & 0.0834 \\
|
|
& & $\Ex{(2)}$ & 0.1785 & 0.1769 & 0.1728 & 0.1652 & 0.1520 & 0.1342 & 0.1063 \\
|
|
& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
|
|
& & $\DD{c}{(1)}$ & 0.0244 & 0.0235 & 0.0218 & 0.0186 & 0.0130 & 0.0043 & -0.0065 \\
|
|
& & $\DD{c}{(2)}$ & 0.0589 & 0.0574 & 0.0546 & 0.0496 & 0.0410 & 0.0275 & 0.0095 \\
|
|
& & $\LZ{c}{}$ & 0.0015 & 0.0029 & 0.0055 & 0.0103 & 0.0180 & 0.0284 & 0.038555 \\
|
|
\\
|
|
eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0014 & 0.0013 & 0.0012 & 0.0009 & 0.0003 & -0.0013 & -0.0079 \\
|
|
& & $\E{(1)}$ & 0.0149 & 0.0138 & 0.0115 & 0.0072 & -0.0001 & -0.0110 & -0.0209 \\
|
|
& & $\E{(2)}$ & 0.1217 & 0.1198 & 0.1154 & 0.1069 & 0.0917 & 0.0691 & 0.0389 \\
|
|
& & $\Ex{(1)}$ & 0.0135 & 0.0125 & 0.0103 & 0.0063 & -0.0005 & -0.0096 & -0.0130 \\
|
|
& & $\Ex{(2)}$ & 0.1203 & 0.1185 & 0.1142 & 0.1060 & 0.0914 & 0.0705 & 0.0469 \\
|
|
& & $\DD{c}{(0)}$ & -0.0278 & -0.0270 & -0.0255 & -0.0227 & -0.0180 & -0.0105 & -0.0007 \\
|
|
& & $\DD{c}{(1)}$ & -0.0034 & -0.0034 & -0.0037 & -0.0041 & -0.0050 & -0.0063 & -0.0076 \\
|
|
& & $\DD{c}{(2)}$ & 0.0311 & 0.0304 & 0.0291 & 0.0268 & 0.0230 & 0.0168 & 0.0083 \\
|
|
& & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0027 & 0.0051 & 0.0094 & 0.0159 & 0.0242 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
|
|
|
|
\bibliography{../eDFT}
|
|
|
|
\end{document}
|